Lectures on Electromagnetic Field Theory
Weng Cho CHEW1
Fall 2020, Purdue University
1Updated: December 3, 2020
Contents
Preface xi
Acknowledgements xii
1 Introduction, Maxwell’s Equations 1 1.1 Importance of Electromagnetics ...... 1 1.1.1 A Brief History of Electromagnetics ...... 4 1.2 Maxwell’s Equations in Integral Form ...... 7 1.3 Static Electromagnetics ...... 8 1.3.1 Coulomb’s Law (Statics) ...... 8 1.3.2 Electric Field E (Statics) ...... 8 1.3.3 Gauss’s Law (Statics) ...... 11 1.3.4 Derivation of Gauss’s Law from Coulomb’s Law (Statics) ...... 12 1.4 Homework Examples ...... 13
2 Maxwell’s Equations, Differential Operator Form 17 2.1 Gauss’s Divergence Theorem ...... 17 2.1.1 Some Details ...... 19 2.1.2 Gauss’s Law in Differential Operator Form ...... 21 2.1.3 Physical Meaning of Divergence Operator ...... 21 2.2 Stokes’s Theorem ...... 22 2.2.1 Faraday’s Law in Differential Operator Form ...... 25 2.2.2 Physical Meaning of Curl Operator ...... 26 2.3 Maxwell’s Equations in Differential Operator Form ...... 26 2.4 Homework Examples ...... 27 2.5 Historical Notes ...... 28
3 Constitutive Relations, Wave Equation, Electrostatics, and Static Green’s Function 29 3.1 Simple Constitutive Relations ...... 29 3.2 Emergence of Wave Phenomenon, Triumph of Maxwell’s Equations ...... 30 3.3 Static Electromagnetics–Revisted ...... 34 3.3.1 Electrostatics ...... 34
i ii Electromagnetic Field Theory
3.3.2 Poisson’s Equation ...... 35 3.3.3 Static Green’s Function ...... 36 3.3.4 Laplace’s Equation ...... 36 3.4 Homework Examples ...... 37
4 Magnetostatics, Boundary Conditions, and Jump Conditions 39 4.1 Magnetostatics ...... 39 4.1.1 More on Coulomb Gauge ...... 41 4.2 Boundary Conditions–1D Poisson’s Equation ...... 41 4.3 Boundary Conditions–Maxwell’s Equations ...... 43 4.3.1 Faraday’s Law ...... 44 4.3.2 Gauss’s Law for Electric Flux ...... 45 4.3.3 Ampere’s Law ...... 46 4.3.4 Gauss’s Law for Magnetic Flux ...... 48
5 Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem 49 5.1 Derivation of Biot-Savart Law ...... 50 5.2 Shielding by Conductive Media ...... 52 5.2.1 Boundary Conditions–Conductive Media Case ...... 52 5.2.2 Electric Field Inside a Conductor ...... 53 5.2.3 Magnetic Field Inside a Conductor ...... 54 5.3 Instantaneous Poynting’s Theorem ...... 56
6 Time-Harmonic Fields, Complex Power 61 6.1 Time-Harmonic Fields—Linear Systems ...... 62 6.2 Fourier Transform Technique ...... 64 6.3 Complex Power ...... 65
7 More on Constitute Relations, Uniform Plane Wave 69 7.1 More on Constitutive Relations ...... 69 7.1.1 Isotropic Frequency Dispersive Media ...... 69 7.1.2 Anisotropic Media ...... 71 7.1.3 Bi-anisotropic Media ...... 72 7.1.4 Inhomogeneous Media ...... 72 7.1.5 Uniaxial and Biaxial Media ...... 72 7.1.6 Nonlinear Media ...... 73 7.2 Wave Phenomenon in the Frequency Domain ...... 73 7.3 Uniform Plane Waves in 3D ...... 75
8 Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 79 8.1 Plane Waves in Lossy Conductive Media ...... 79 8.1.1 Highly Conductive Case ...... 80 8.1.2 Lowly Conductive Case ...... 81 8.2 Lorentz Force Law ...... 82 8.3 Drude-Lorentz-Sommerfeld Model ...... 82 Contents iii
8.3.1 Cold Collisionless Plasma Medium ...... 83 8.3.2 Bound Electron Case ...... 85 8.3.3 Damping or Dissipation Case ...... 85 8.3.4 Broad Applicability of Drude-Lorentz-Sommerfeld Model ...... 86 8.3.5 Frequency Dispersive Media ...... 88 8.3.6 Plasmonic Nanoparticles ...... 89
9 Waves in Gyrotropic Media, Polarization 91 9.1 Gyrotropic Media and Faraday Rotation ...... 91 9.2 Wave Polarization ...... 94 9.2.1 Arbitrary Polarization Case and Axial Ratio1 ...... 96 9.3 Polarization and Power Flow ...... 98
10 Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condi- tion, Energy Density 101 10.1 Spin Angular Momentum and Cylindrical Vector Beam ...... 102 10.2 Momentum Density of Electromagnetic Field ...... 103 10.3 Complex Poynting’s Theorem and Lossless Conditions ...... 104 10.3.1 Complex Poynting’s Theorem ...... 104 10.3.2 Lossless Conditions ...... 105 10.4 Energy Density in Dispersive Media ...... 107
11 Transmission Lines 111 11.1 Transmission Line Theory ...... 112 11.1.1 Time-Domain Analysis ...... 113 11.1.2 Frequency-Domain Analysis–the Power of Phasor Technique Again! . . 116 11.2 Lossy Transmission Line ...... 117
12 More on Transmission Lines 121 12.1 Terminated Transmission Lines ...... 121 12.1.1 Shorted Terminations ...... 124 12.1.2 Open Terminations ...... 125 12.2 Smith Chart ...... 126 12.3 VSWR (Voltage Standing Wave Ratio) ...... 128
13 Multi-Junction Transmission Lines, Duality Principle 133 13.1 Multi-Junction Transmission Lines ...... 133 13.1.1 Single-Junction Transmission Lines ...... 135 13.1.2 Two-Junction Transmission Lines ...... 136 13.1.3 Stray Capacitance and Inductance ...... 139 13.2 Duality Principle ...... 140 13.2.1 Unusual Swaps2 ...... 141 13.3 Fictitious Magnetic Currents ...... 142
1This section is mathematically complicated. It can be skipped on first reading. 2This section can be skipped on first reading. iv Electromagnetic Field Theory
14 Reflection, Transmission, and Interesting Physical Phenomena 145 14.1 Reflection and Transmission—Single Interface Case ...... 145 14.1.1 TE Polarization (Perpendicular or E Polarization)3 ...... 146 14.1.2 TM Polarization (Parallel or H Polarization)4 ...... 148 14.2 Interesting Physical Phenomena ...... 149 14.2.1 Total Internal Reflection ...... 150
15 More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines 155 15.1 Brewster’s Angle ...... 155 15.1.1 Surface Plasmon Polariton ...... 158 15.2 Homomorphism of Uniform Plane Waves and Transmission Lines Equations . 160 15.2.1 TE or TEz Waves ...... 161 15.2.2 TM or TMz Waves ...... 162
16 Waves in Layered Media 165 16.1 Waves in Layered Media ...... 165 16.1.1 Generalized Reflection Coefficient for Layered Media ...... 166 16.1.2 Ray Series Interpretation of Generalized Reflection Coefficient . . . . 167 16.1.3 Guided Modes from Generalized Reflection Coefficients ...... 168 16.2 Phase Velocity and Group Velocity ...... 168 16.2.1 Phase Velocity ...... 168 16.2.2 Group Velocity ...... 170 16.3 Wave Guidance in a Layered Media ...... 173 16.3.1 Transverse Resonance Condition ...... 173
17 Dielectric Waveguides 175 17.1 Generalized Transverse Resonance Condition ...... 175 17.2 Dielectric Waveguide ...... 176 17.2.1 TE Case ...... 177 17.2.2 TM Case ...... 183 17.2.3 A Note on Cut-Off of Dielectric Waveguides ...... 184
18 Hollow Waveguides 185 18.1 General Information on Hollow Waveguides ...... 185 18.1.1 Absence of TEM Mode in a Hollow Waveguide ...... 186 18.1.2 TE Case (Ez = 0, Hz 6= 0, TEz case) ...... 187 18.1.3 TM Case (Ez 6= 0, Hz = 0, TMz Case) ...... 189 18.2 Rectangular Waveguides ...... 190 18.2.1 TE Modes (Hz 6= 0, H Modes or TEz Modes) ...... 190
3 These polarizations are also variously know as TEz, or the s and p polarizations, a descendent from the notations for acoustic waves where s and p stand for shear and pressure waves respectively. 4 Also known as TMz polarization. Contents v
19 More on Hollow Waveguides 193 19.1 Rectangular Waveguides, Contd...... 194 19.1.1 TM Modes (Ez 6= 0, E Modes or TMz Modes) ...... 194 19.1.2 Bouncing Wave Picture ...... 195 19.1.3 Field Plots ...... 196 19.2 Circular Waveguides ...... 198 19.2.1 TE Case ...... 198 19.2.2 TM Case ...... 200
20 More on Waveguides and Transmission Lines 205 20.1 Circular Waveguides, Contd...... 205 20.1.1 An Application of Circular Waveguide ...... 206 20.2 Remarks on Quasi-TEM Modes, Hybrid Modes, and Surface Plasmonic Modes 209 20.2.1 Quasi-TEM Modes ...... 209 20.2.2 Hybrid Modes–Inhomogeneously-Filled Waveguides ...... 210 20.2.3 Guidance of Modes ...... 211 20.3 Homomorphism of Waveguides and Transmission Lines ...... 212 20.3.1 TE Case ...... 212 20.3.2 TM Case ...... 214 20.3.3 Mode Conversion ...... 216
21 Cavity Resonators 219 21.1 Transmission Line Model of a Resonator ...... 219 21.2 Cylindrical Waveguide Resonators ...... 221 21.2.1 βz = 0 Case ...... 223 21.2.2 Lowest Mode of a Rectangular Cavity ...... 224 21.3 Some Applications of Resonators ...... 225 21.3.1 Filters ...... 226 21.3.2 Electromagnetic Sources ...... 227 21.3.3 Frequency Sensor ...... 230
22 Quality Factor of Cavities, Mode Orthogonality 233 22.1 The Quality Factor of a Cavity–General Concept ...... 233 22.1.1 Analogue with an LC Tank Circuit ...... 234 22.1.2 Relation to Bandwidth and Pole Location ...... 236 22.1.3 Wall Loss and Q for a Metallic Cavity ...... 237 22.1.4 Example: The Q of TM110 Mode ...... 239 22.2 Mode Orthogonality and Matrix Eigenvalue Problem ...... 240 22.2.1 Matrix Eigenvalue Problem (EVP) ...... 240 22.2.2 Homomorphism with the Waveguide Mode Problem ...... 241 22.2.3 Proof of Orthogonality of Waveguide Modes5 ...... 242
5This may be skipped on first reading. vi Electromagnetic Field Theory
23 Scalar and Vector Potentials 245 23.1 Scalar and Vector Potentials for Time-Harmonic Fields ...... 245 23.2 Scalar and Vector Potentials for Statics, A Review ...... 246 23.2.1 Scalar and Vector Potentials for Electrodynamics ...... 247 23.2.2 More on Scalar and Vector Potentials ...... 249 23.3 When is Static Electromagnetic Theory Valid? ...... 250 23.3.1 Quasi-Static Electromagnetic Theory ...... 255
24 Circuit Theory Revisited 257 24.1 Kirchhoff Current Law ...... 257 24.2 Kirchhoff Voltage Law ...... 258 24.3 Inductor ...... 262 24.4 Capacitance ...... 263 24.5 Resistor ...... 263 24.6 Some Remarks ...... 264 24.7 Energy Storage Method for Inductor and Capacitor ...... 264 24.8 Finding Closed-Form Formulas for Inductance and Capacitance ...... 265 24.9 Importance of Circuit Theory in IC Design ...... 267 24.10Decoupling Capacitors and Spiral Inductors ...... 269
25 Radiation by a Hertzian Dipole 271 25.1 History ...... 271 25.2 Approximation by a Point Source ...... 273 25.2.1 Case I. Near Field, βr 1 ...... 275 25.2.2 Case II. Far Field (Radiation Field), βr 1 ...... 276 25.3 Radiation, Power, and Directive Gain Patterns ...... 276 25.3.1 Radiation Resistance ...... 279
26 Radiation Fields 283 26.1 Radiation Fields or Far-Field Approximation ...... 284 26.1.1 Far-Field Approximation ...... 285 26.1.2 Locally Plane Wave Approximation ...... 286 26.1.3 Directive Gain Pattern Revisited ...... 289
27 Array Antennas, Fresnel Zone, Rayleigh Distance 293 27.1 Linear Array of Dipole Antennas ...... 293 27.1.1 Far-Field Approximation ...... 294 27.1.2 Radiation Pattern of an Array ...... 295 27.2 When is Far-Field Approximation Valid? ...... 298 27.2.1 Rayleigh Distance ...... 299 27.2.2 Near Zone, Fresnel Zone, and Far Zone ...... 300 Contents vii
28 Different Types of Antennas—Heuristics 303 28.1 Resonance Tunneling in Antenna ...... 304 28.2 Horn Antennas ...... 307 28.3 Quasi-Optical Antennas ...... 309 28.4 Small Antennas ...... 311
29 Uniqueness Theorem 317 29.1 The Difference Solutions to Source-Free Maxwell’s Equations ...... 317 29.2 Conditions for Uniqueness ...... 320 29.2.1 Isotropic Case ...... 320 29.2.2 General Anisotropic Case ...... 321 29.3 Hind Sight Using Linear Algebra ...... 322 29.4 Connection to Poles of a Linear System ...... 323 29.5 Radiation from Antenna Sources and Radiation Condition ...... 324
30 Reciprocity Theorem 327 30.1 Mathematical Derivation ...... 328 30.2 Conditions for Reciprocity ...... 331 30.3 Application to a Two-Port Network and Circuit Theory ...... 331 30.4 Voltage Sources in Electromagnetics ...... 333 30.5 Hind Sight ...... 334 30.6 Transmit and Receive Patterns of an Antennna ...... 335
31 Equivalence Theorems, Huygens’ Principle 339 31.1 Equivalence Theorems or Equivalence Principles ...... 339 31.1.1 Inside-Out Case ...... 340 31.1.2 Outside-in Case ...... 341 31.1.3 General Case ...... 341 31.2 Electric Current on a PEC ...... 342 31.3 Magnetic Current on a PMC ...... 343 31.4 Huygens’ Principle and Green’s Theorem ...... 343 31.4.1 Scalar Waves Case ...... 344 31.4.2 Electromagnetic Waves Case ...... 346
32 Shielding, Image Theory 349 32.1 Shielding ...... 349 32.1.1 A Note on Electrostatic Shielding ...... 349 32.1.2 Relaxation Time ...... 350 32.2 Image Theory ...... 351 32.2.1 Electric Charges and Electric Dipoles ...... 352 32.2.2 Magnetic Charges and Magnetic Dipoles ...... 353 32.2.3 Perfect Magnetic Conductor (PMC) Surfaces ...... 354 32.2.4 Multiple Images ...... 355 32.2.5 Some Special Cases—Spheres, Cylinders, and Dielectric Interfaces . . 356 viii Electromagnetic Field Theory
33 High Frequency Solutions, Gaussian Beams 361 33.1 Tangent Plane Approximations ...... 362 33.2 Fermat’s Principle ...... 363 33.2.1 Generalized Snell’s Law ...... 365 33.3 Gaussian Beam ...... 366 33.3.1 Derivation of the Paraxial/Parabolic Wave Equation ...... 366 33.3.2 Finding a Closed Form Solution ...... 367 33.3.3 Other solutions ...... 369
34 Scattering of Electromagnetic Field 371 34.1 Rayleigh Scattering ...... 371 34.1.1 Scattering by a Small Spherical Particle ...... 373 34.1.2 Scattering Cross Section ...... 375 34.1.3 Small Conductive Particle ...... 378 34.2 Mie Scattering ...... 379 34.2.1 Optical Theorem ...... 380 34.2.2 Mie Scattering by Spherical Harmonic Expansions ...... 381 34.2.3 Separation of Variables in Spherical Coordinates6 ...... 381
35 Spectral Expansions of Source Fields—Sommerfeld Integrals 383 35.1 Spectral Representations of Sources ...... 383 35.1.1 A Point Source ...... 384 35.2 A Source on Top of a Layered Medium ...... 389 35.2.1 Electric Dipole Fields–Spectral Expansion ...... 389 35.3 Stationary Phase Method—Fermat’s Principle ...... 392 35.4 Riemann Sheets and Branch Cuts7 ...... 396 35.5 Some Remarks8 ...... 396
36 Computational Electromagnetics, Numerical Methods 399 36.1 Computational Electromagnetics, Numerical Methods ...... 401 36.2 Examples of Differential Equations ...... 401 36.3 Examples of Integral Equations ...... 402 36.3.1 Volume Integral Equation ...... 402 36.3.2 Surface Integral Equation ...... 404 36.4 Function as a Vector ...... 405 36.5 Operator as a Map ...... 406 36.5.1 Domain and Range Spaces ...... 406 36.6 Approximating Operator Equations with Matrix Equations ...... 407 36.6.1 Subspace Projection Methods ...... 407 36.6.2 Dual Spaces ...... 408 36.6.3 Matrix and Vector Representations ...... 408 36.6.4 Mesh Generation ...... 409
6May be skipped on first reading. 7This may be skipped on first reading. 8This may be skipped on first reading. Contents ix
36.6.5 Differential Equation Solvers versus Integral Equation Solvers . . . . . 410 36.7 Solving Matrix Equation by Optimization ...... 410 36.7.1 Gradient of a Functional ...... 411
37 Finite Difference Method, Yee Algorithm 415 37.1 Finite-Difference Time-Domain Method ...... 415 37.1.1 The Finite-Difference Approximation ...... 416 37.1.2 Time Stepping or Time Marching ...... 418 37.1.3 Stability Analysis ...... 420 37.1.4 Grid-Dispersion Error ...... 422 37.2 The Yee Algorithm ...... 424 37.2.1 Finite-Difference Frequency Domain Method ...... 427 37.3 Absorbing Boundary Conditions ...... 428
38 Quantum Theory of Light 431 38.1 Historical Background on Quantum Theory ...... 431 38.2 Connecting Electromagnetic Oscillation to Simple Pendulum ...... 434 38.3 Hamiltonian Mechanics ...... 438 38.4 Schr¨odingerEquation (1925) ...... 440 38.5 Some Quantum Interpretations–A Preview ...... 443 38.5.1 Matrix or Operator Representations ...... 444 38.6 Bizarre Nature of the Photon Number States ...... 445
39 Quantum Coherent State of Light 447 39.1 The Quantum Coherent State ...... 447 39.1.1 Quantum Harmonic Oscillator Revisited ...... 448 39.2 Some Words on Quantum Randomness and Quantum Observables ...... 450 39.3 Derivation of the Coherent States ...... 451 39.3.1 Time Evolution of a Quantum State ...... 453 39.4 More on the Creation and Annihilation Operator ...... 454 39.4.1 Connecting Quantum Pendulum to Electromagnetic Oscillator9 . . . . 457 39.5 Epilogue ...... 460
9May be skipped on first reading. x Electromagnetic Field Theory Preface
This set of lecture notes is from my teaching of ECE 604, Electromagnetic Field Theory, at ECE, Purdue University, West Lafayette. It is intended for entry level graduate students. Because different universities have different undergraduate requirements in electromagnetic field theory, this is a course intended to “level the playing field”. From this point onward, hopefully, all students will have the fundamental background in electromagnetic field theory needed to take advance level courses and do research at Purdue. In developing this course, I have drawn heavily upon knowledge of our predecessors in this area. Many of the textbooks and papers used, I have listed them in the reference list. Being a practitioner in this field for over 40 years, I have seen electromagnetic theory im- pacting modern technology development unabated. Despite its age, the set of Maxwell’s equations has endured and continued to be important, from statics to optics, from classi- cal to quantum, and from nanometer lengthscales to galactic lengthscales. The applications of electromagnetic technologies have also been tremendous and wide-ranging: from geophysi- cal exploration, remote sensing, bio-sensing, electrical machinery, renewable and clean energy, biomedical engineering, optics and photonics, computer chip, computer system, and quantum computer designs, quantum communication and many more. Electromagnetic field theory is not everything, but is remains an important component of modern technology developments. The challenge in teaching this course is on how to teach over 150 years of knowledge in one semester: Of course this is mission impossible! To do this, we use the traditional wisdom of engineering education: Distill the knowledge, make it as simple as possible, and teach the fundamental big ideas in one short semester. Because of this, you may find the flow of the lectures erratic. Some times, I feel the need to touch on certain big ideas before moving on, resulting in the choppiness of the curriculum. Also, in this course, I exploit mathematical homomorphism as much as possible to simplify the teaching. After years of practising in this area, I find that some complex and advanced concepts become simpler if mathematical homomorphism is exploited between the advanced concepts and simpler ones. An example of this is on waves in layered media. The problem is homomorphic to the transmission line problem: Hence, using transmission line theory, one can simplify the derivations of some complicated formulas. A large part of modern electromagnetic technologies is based on heuristics. This is some- thing difficult to teach, as it relies on physical insight and experience. Modern commercial software has reshaped this landscape, as the field of math-physics modeling through numer- ical simulations, known as computational electromagnetic (CEM), has made rapid advances in recent years. Many cut-and-try laboratory experiments, based on heuristics, have been
xi xii Lectures on Electromagnetic Field Theory replaced by cut-and-try computer experiments, which are a lot cheaper. An exciting modern development is the role of electromagnetics and Maxwell’s equations in quantum technologies. We will connect Maxwell’s equations to them toward the end of this course. This is a challenge, as it has never been done before at an entry level course to my knowledge.
Weng Cho CHEW December 3, 2020 Purdue University
Acknowledgements I like to thank Dan Jiao for sharing her lecture notes in this course, as well as Andy Weiner for sharing his experience in teaching this course. Mark Lundstrom gave me useful feedback on Lectures 38 and 39 on the quantum theory of light. Also, I am thankful to Dong-Yeop Na for helping teach part of this course. Robert Hsueh-Yung Chao also took time to read the lecture notes and gave me very useful feedback. Lecture 1
Introduction, Maxwell’s Equations
In the beginning, this field is either known as electricity and magnetism or optics. But later, as we shall discuss, these two fields are found to be based on the same set equations known as Maxwell’s equations. Maxwell’s equations unified these two fields, and it is common to call the study of electromagnetic theory based on Maxwell’s equations electromagnetics. It has wide-ranging applications from statics to ultra-violet light in the present world with impact on many different technologies.
1.1 Importance of Electromagnetics
We will explain why electromagnetics is so important, and its impact on very many different areas. Then we will give a brief history of electromagnetics, and how it has evolved in the modern world. Next we will go briefly over Maxwell’s equations in their full glory. But we will begin the study of electromagnetics by focussing on static problems which are valid in the long-wavelength limit. It has been based on Maxwell’s equations, which are the result of the seminal work of James Clerk Maxwell completed in 1865, after his presentation to the British Royal Society in 1864. It has been over 150 years ago now, and this is a long time compared to the leaps and bounds progress we have made in technological advancements. Nevertheless, research in electromagnetics has continued unabated despite its age. The reason is that electromagnetics is extremely useful, and has impacted a large sector of modern technologies. To understand why electromagnetics is so useful, we have to understand a few points about Maxwell’s equations. Maxwell’s equations are valid over a vast length scale from subatomic dimensions to galactic dimensions. Hence, these equations are valid over a vast range of wavelengths, going from static to ultra-violet wavelengths.1
1Current lithography process is working with using ultra-violet light with a wavelength of 193 nm.
1 2 Electromagnetic Field Theory
Maxwell’s equations are relativistic invariant in the parlance of special relativity [1]. In fact, Einstein was motivated with the theory of special relativity in 1905 by Maxwell’s equations [2]. These equations look the same, irrespective of what inertial reference frame one is in.
Maxwell’s equations are valid in the quantum regime, as it was demonstrated by Paul Dirac in 1927 [3]. Hence, many methods of calculating the response of a medium to classical field can be applied in the quantum regime also. When electromagnetic theory is combined with quantum theory, the field of quantum optics came about. Roy Glauber won a Nobel prize in 2005 because of his work in this area [4].
Maxwell’s equations and the pertinent gauge theory has inspired Yang-Mills theory (1954) [5], which is also known as a generalized electromagnetic theory. Yang-Mills theory is motivated by differential forms in differential geometry [6]. To quote from Misner, Thorne, and Wheeler, “Differential forms illuminate electromagnetic theory, and electromagnetic theory illuminates differential forms.” [7, 8]
Maxwell’s equations are some of the most accurate physical equations that have been validated by experiments. In 1985, Richard Feynman wrote that electromagnetic theory had been validated to one part in a billion.2 Now, it has been validated to one part in a trillion (Aoyama et al, Styer, 2012).3
As a consequence, electromagnetics has had a tremendous impact in science and tech- nology. This is manifested in electrical engineering, optics, wireless and optical commu- nications, computers, remote sensing, bio-medical engineering etc.
2This means that if a jet is to fly from New York to Los Angeles, an error of one part in a billion means an error of a few millmeters. 3This means an error of a hairline, if one were to fly from the earth to the moon. Introduction, Maxwell’s Equations 3
Figure 1.1: The impact of electromagnetics in many technologies. The areas in blue are prevalent areas impacted by electromagnetics some 20 years ago [9], and the areas in brown are modern emerging areas impacted by electromagnetics. 4 Electromagnetic Field Theory
Figure 1.2: Knowledge grows like a tree. Engineering knowledge and real-world applica- tions are driven by fundamental knowledge from math and sciences. At a university, we do science-based engineering research that can impact wide-ranging real-world applications. But everyone is equally important in transforming our society. Just like the parts of the human body, no one can claim that one is more important than the others.
Figure 1.2 shows how knowledge are driven by basic math and science knowledge. Its growth is like a tree. It is important that we collaborate to develop technologies that can transform this world.
1.1.1 A Brief History of Electromagnetics Electricity and magnetism have been known to mankind for a long time. Also, the physical properties of light have been known. But electricity and magnetism, now termed electromag- netics in the modern world, has been thought to be governed by different physical laws as opposed to optics. This is understandable as the physics of electricity and magnetism is quite different of the physics of optics as they were known to humans then. For example, lode stone was known to the ancient Greek and Chinese around 600 BC to 400 BC. Compass was used in China since 200 BC. Static electricity was reported by the Greek as early as 400 BC. But these curiosities did not make an impact until the age of telegraphy. The coming about of telegraphy was due to the invention of the voltaic cell or the galvanic cell in the late 1700’s, by Luigi Galvani and Alesandro Volta [10]. It was soon discovered that two pieces of wire, connected to a voltaic cell, can transmit information. Introduction, Maxwell’s Equations 5
So by the early 1800’s this possibility had spurred the development of telegraphy. Both Andr´e-MarieAmp´ere(1823) [11, 12] and Michael Faraday (1838) [13] did experiments to better understand the properties of electricity and magnetism. And hence, Ampere’s law and Faraday law are named after them. Kirchhoff voltage and current laws were also developed in 1845 to help better understand telegraphy [14, 15]. Despite these laws, the technology of telegraphy was poorly understood. For instance, it was not known as to why the telegraphy signal was distorted. Ideally, the signal should be a digital signal switching between one’s and zero’s, but the digital signal lost its shape rapidly along a telegraphy line.4
It was not until 1865 that James Clerk Maxwell [17] put in the missing term in Am- pere’s law, the displacement current term, only then the mathematical theory for electricity and magnetism was complete. Ampere’s law is now known as generalized Ampere’s law. The complete set of equations are now named Maxwell’s equations in honor of James Clerk Maxwell.
The rousing success of Maxwell’s theory was that it predicted wave phenomena, as they have been observed along telegraphy lines. But it was not until 23 years later that Heinrich Hertz in 1888 [18] did experiment to prove that electromagnetic field can propagate through space across a room. This illustrates the difficulty of knowledge dissemination when new knowledge is discovered. Moreover, from experimental measurement of the permittivity and permeability of matter, it was decided that electromagnetic wave moves at a tremendous speed. But the velocity of light has been known for a long while from astronomical obser- vations (Roemer, 1676) [19]. The observation of interference phenomena in light has been known as well. When these pieces of information were pieced together, it was decided that electricity and magnetism, and optics, are actually governed by the same physical law or Maxwell’s equations. And optics and electromagnetics are unified into one field!
4As a side note, in 1837, Morse invented the Morse code for telegraphy [16]. There were cross pollination of ideas across the Atlantic ocean despite the distance. In fact, Benjamin Franklin associated lightning with electricity in the latter part of the 18-th century. Also, notice that electrical machinery was invented in 1832 even though electromagnetic theory was not fully understood. 6 Electromagnetic Field Theory
Figure 1.3: A brief history of electromagnetics and optics as depicted in this figure.
In Figure 1.3, a brief history of electromagnetics and optics is depicted. In the begin- ning, it was thought that electricity and magnetism, and optics were governed by different physical laws. Low frequency electromagnetics was governed by the understanding of fields and their interaction with media. Optical phenomena were governed by ray optics, reflection and refraction of light. But the advent of Maxwell’s equations in 1865 revealed that they can be unified under electromagnetic theory. Then solving Maxwell’s equations becomes a mathematical endeavor. The photo-electric effect [20,21], and Planck radiation law [22] point to the fact that elec- tromagnetic energy is manifested in terms of packets of energy, indicating the corpuscular nature of light. Each unit of this energy is now known as the photon. A photon carries an en- ergy packet equal to ~ω, where ω is the angular frequency of the photon and ~ = 6.626×10−34 J s, the Planck constant, which is a very small constant. Hence, the higher the frequency, the easier it is to detect this packet of energy, or feel the graininess of electromagnetic en- ergy. Eventually, in 1927 [3], quantum theory was incorporated into electromagnetics, and the quantum nature of light gives rise to the field of quantum optics. Recently, even microwave photons have been measured [23,24]. They are difficult to detect because of the low frequency of microwave (109 Hz) compared to optics (1015 Hz): a microwave photon carries a packet of energy about a million times smaller than that of optical photon. The progress in nano-fabrication [25] allows one to make optical components that are subwavelength as the wavelength of blue light is about 450 nm. As a result, interaction of light with nano-scale optical components requires the solution of Maxwell’s equations in its full glory. Introduction, Maxwell’s Equations 7
In the early days of quantum theory, there were two prevailing theories of quantum in- terpretation. Quantum measurements were found to be random. In order to explain the probabilistic nature of quantum measurements, Einstein posited that a random hidden vari- able controlled the outcome of an experiment. On the other hand, the Copenhagen school of interpretation led by Niels Bohr, asserted that the outcome of a quantum measurement is not known until after a measurement [26]. In 1960s, Bell’s theorem (by John Steward Bell) [27] said that an inequality should be satisfied if Einstein’s hidden variable theory was correct. Otherwise, the Copenhagen school of interpretation should prevail. However, experimental measurement showed that the in- equality was violated, favoring the Copenhagen school of quantum interpretation [26]. This interpretation says that a quantum state is in a linear superposition of states before a mea- surement. But after a measurement, a quantum state collapses to the state that is measured. This implies that quantum information can be hidden incognito in a quantum state. Hence, a quantum particle, such as a photon, its state is unknown until after its measurement. In other words, quantum theory is “spooky”. This leads to growing interest in quantum infor- mation and quantum communication using photons. Quantum technology with the use of photons, an electromagnetic quantum particle, is a subject of growing interest. This also has the profound and beautiful implication that “our karma is not written on our forehead when we were born, our future is in our own hands!”
1.2 Maxwell’s Equations in Integral Form
Maxwell’s equations can be presented as fundamental postulates.5 We will present them in their integral forms, but will not belabor them until later. ¤ d E · dl = − B · dS Faraday’s Law (1.2.1) C dt¤ S d H · dl = D · dS + I Ampere’s Law (1.2.2) C dt S D · dS = Q Gauss’s or Coulomb’s Law (1.2.3) S B · dS = 0 Gauss’s Law (1.2.4) S The units of the basic quantities above are given as: E: V/m H: A/m D: C/m2 B: W/m2 I:A Q:C where V=volts, A=amperes, C=coulombs, and W=webers. 5Postulates in physics are similar to axioms in mathematics. They are assumptions that need not be proved. 8 Electromagnetic Field Theory 1.3 Static Electromagnetics 1.3.1 Coulomb’s Law (Statics)
This law, developed in 1785 [28], expresses the force between two charges q1 and q2. If these charges are positive, the force is repulsive and it is given by
q1q2 f = ˆr (1.3.1) 1→2 4πεr2 12
Figure 1.4: The force between two charges q1 and q2. The force is repulsive if the two charges have the same sign. where the units are: f (force): newton q (charge): coulombs ε (permittivity): farads/meter r (distance between q1 and q2): m ˆr12= unit vector pointing from charge 1 to charge 2
r2 − r1 ˆr12 = , r = |r2 − r1| (1.3.2) |r2 − r1| Since the unit vector can be defined in the above, the force between two charges can also be rewritten as
q1q2(r2 − r1) f1→2 = 3 , (r1, r2 are position vectors) (1.3.3) 4πε|r2 − r1|
1.3.2 Electric Field E (Statics) The electric field E is defined as the force per unit charge [29]. For two charges, one of charge q and the other one of incremental charge ∆q, the force between the two charges, according to Coulomb’s law (1.3.1), is q∆q f = ˆr (1.3.4) 4πεr2 Introduction, Maxwell’s Equations 9 where ˆr is a unit vector pointing from charge q to the incremental charge ∆q. Then the electric field E, which is the force per unit charge, is given by f E = , (V/m) (1.3.5) 4q
This electric field E from a point charge q at the orgin is hence q E = ˆr (1.3.6) 4πεr2 Therefore, in general, the electric field E(r) at location r from a point charge q at r0 is given by q(r − r0) E(r) = (1.3.7) 4πε|r − r0|3 where the unit vector r − r0 ˆr = (1.3.8) |r − r0|
Figure 1.5: Emanating E field from an electric point charge as depicted by depicted by (1.3.7) and (1.3.6).
If one knows E due to a point charge, one will know E due to any charge distribution because any charge distribution can be decomposed into sum of point charges. For instance, if 10 Electromagnetic Field Theory
there are N point charges each with amplitude qi, then by the principle of linear superposition, the total field produced by these N charges is
N X qi(r − ri) E(r) = (1.3.9) 4πε|r − r |3 i=1 i where qi = %(ri)∆Vi is the incremental charge at ri enclosed in the volume ∆Vi. In the continuum limit, one gets ¢ %(r0)(r − r0) E(r) = 0 3 dV (1.3.10) V 4πε|r − r | In other words, the total field, by the principle of linear superposition, is the integral sum- mation of the contributions from the distributed charge density %(r). Introduction, Maxwell’s Equations 11
1.3.3 Gauss’s Law (Statics)
This law is also known as Coulomb’s law as they are closely related to each other. Apparently, this simple law was first expressed by Joseph Louis Lagrange [30] and later, reexpressed by Gauss in 1813 (Wikipedia). This law can be expressed as D · dS = Q (1.3.11) S where D: electric flux density with unit C/m2 and D = εE. dS: an incremental surface at the point on S given by dSnˆ where nˆ is the unit normal pointing outward away from the surface. Q: total charge enclosed by the surface S.
Figure 1.6: Electric flux (courtesy of Ramo, Whinnery, and Van Duzer [31]).
The left-hand side of (1.3.11) represents a surface integral over a closed surface S. To understand it, one can break the surface into a sum of incremental surfaces ∆Si, with a local unit normal nˆi associated with it. The surface integral can then be approximated by a summation X X D · dS ≈ Di · nˆi∆Si = Di · ∆Si (1.3.12) S i i where one has defined ∆Si = nˆi∆Si. In the limit when ∆Si becomes infinitesimally small, the summation becomes a surface integral. 12 Electromagnetic Field Theory
1.3.4 Derivation of Gauss’s Law from Coulomb’s Law (Statics) From Coulomb’s law and the ensuing electric field due to a point charge, the electric flux is q D = εE = ˆr (1.3.13) 4πr2 When a closed spherical surface S is drawn around the point charge q, by symmetry, the electric flux though every point of the surface is the same. Moreover, the normal vector nˆ on the surface is just ˆr. Consequently, D · nˆ = D · ˆr = q/(4πr2), which is a constant on a spherical of radius r. Hence, we conclude that for a point charge q, and the pertinent electric flux D that it produces on a spherical surface, 2 2 D · dS = 4πr D · nˆ = 4πr Dr = q (1.3.14) S Therefore, Gauss’s law is satisfied by a point charge.
Figure 1.7: Electric flux from a point charge satisfies Gauss’s law.
Even when the shape of the spherical surface S changes from a sphere to an arbitrary shape surface S, it can be shown that the total flux through S is still q. In other words, the total flux through sufaces S1 and S2 in Figure 1.8 are the same. This can be appreciated by taking a sliver of the angular sector as shown in Figure 1.9. Here, ∆S1 and ∆S2 are two incremental surfaces intercepted by this sliver of angular sector. The amount of flux passing through this incremental surface is given by dS · D = nˆ · D∆S = nˆ · ˆrDr∆S. Here, D = ˆrDr is pointing in the ˆr direction. In ∆S1,n ˆ is pointing in the ˆr direction. But in ∆S2, the incremental area has been enlarged by that nˆ not aligned with D. But this enlargement is compensated by nˆ · ˆr. Also, ∆S2 has grown bigger, but the flux 2 at ∆S2 has grown weaker by the ratio of (r2/r1) . Finally, the two fluxes are equal in the limit that the sliver of angular sector becomes infinitesimally small. This proves the assertion that the total fluxes through S1 and S2 are equal. Since the total flux from a point charge q through a closed surface is independent of its shape, but always equal to q, then if we have a total charge Q which can be expressed as the sum of point charges, namely. X Q = qi (1.3.15) i Introduction, Maxwell’s Equations 13
Figure 1.8: Same amount of electric flux from a point charge passes through two surfaces S1 and S2.
Then the total flux through a closed surface equals the total charge enclosed by it, which is the statement of Gauss’s law or Coulomb’s law.
1.4 Homework Examples
Example 1
Field of a ring of charge of density %l C/m. Question: What is E along z axis? Hint: Use symmetry.
Example 2 Field between coaxial cylinders of unit length. Question: What is E? Hint: Use symmetry and cylindrical coordinates to express E =ρE ˆ ρ and appply Gauss’s law.
Example 3 Fields of a sphere of uniform charge density. Question: What is E? Hint: Again, use symmetry and spherical coordinates to express E =rE ˆ r and appply Gauss’s law. 14 Electromagnetic Field Theory
Figure 1.9: When a sliver of angular sector is taken, same amount of electric flux from a point charge passes through two incremental surfaces ∆S1 and ∆S2.
Figure 1.10: Electric field of a ring of charge (courtesy of Ramo, Whinnery, and Van Duzer [31]).
Figure 1.11: Figure for Example 2 for a coaxial cylinder. Introduction, Maxwell’s Equations 15
Figure 1.12: Figure for Example 3 for a sphere with uniform charge density. 16 Electromagnetic Field Theory Lecture 2
Maxwell’s Equations, Differential Operator Form
Maxwell’s equations were originally written in integral forms as has been shown in the previous lecture. Integral forms have nice physical meaning and can be easily related to experimental measurements. However, the differential operator forms1 can be easily converted to differential equations or partial differential equations where a whole sleuth of mathematical methods and numerical methods can be deployed. Therefore, it is prudent to derive the differential operator form of Maxwell’s equations.
2.1 Gauss’s Divergence Theorem
The divergence theorem is one of the most important theorems in vector calculus [31–34]. First, we will need to prove Gauss’s divergence theorem, namely, that: ¦ dV ∇ · D = D · dS (2.1.1) V S In the above, ∇ · D is defined as D · dS ∇ · D = lim ∆S (2.1.2) ∆V →0 ∆V The above implies that the divergence of the electric flux D, or ∇·D is given by first computing the flux coming (or oozing) out of a small volume ∆V surrounded by a small surface ∆S and taking their ratio as shown on the right-hand side. As shall be shown, the ratio has a limit and eventually, we will find an expression for it. We know that if ∆V ≈ 0 or small, then the
1We caution ourselves not to use the term “differential forms” which has a different meaning used in differential geometry for another form of Maxwell’s equations.
17 18 Electromagnetic Field Theory above,
∆V ∇ · D ≈ D · dS (2.1.3) ∆S
First, we assume that a volume V has been discretized2 into a sum of small cuboids, where the i-th cuboid has a volume of ∆Vi as shown in Figure 2.1. Then
N X V ≈ ∆Vi (2.1.4) i=1
Figure 2.1: The discretization of a volume V into a sum of small volumes ∆Vi each of which is a small cuboid. Stair-casing error occurs near the boundary of the volume V but the error diminishes as ∆Vi → 0.
2Other terms used are “tesselated”, “meshed”, or “gridded”. Maxwell’s Equations, Differential Operator Form 19
Figure 2.2: Fluxes from adjacent cuboids cancel each other leaving only the fluxes at the boundary that remain uncancelled. Please imagine that there is a third dimension of the cuboids in this picture where it comes out of the paper.
Then from (2.1.2),
∆Vi∇ · Di ≈ Di · dSi (2.1.5) ∆Si
By summing the above over all the cuboids, or over i, one gets X X ∆Vi∇ · Di ≈ Di · dSi ≈ D · dS (2.1.6) i i ∆Si S
It is easily seen that the fluxes out of the inner surfaces of the cuboids cancel each other, leaving only fluxes flowing out of the cuboids at the edge of the volume V as explained in Figure 2.2. The right-hand side of the above equation (2.1.6) becomes a surface integral over the surface S except for the stair-casing approximation (see Figure 2.1). However, this approximation becomes increasingly good as ∆Vi → 0. Moreover, the left-hand side becomes a volume integral, and we have ¦ dV∇ · D = D · dS (2.1.7) V S The above is Gauss’s divergence theorem.
2.1.1 Some Details Next, we will derive the details of the definition embodied in (2.1.2). To this end, we evaluate the numerator of the right-hand side carefully, in accordance to Figure 2.3. 20 Electromagnetic Field Theory
Figure 2.3: Figure to illustrate the calculation of fluxes from a small cuboid where a corner of the cuboid is located at (x0, y0, z0). There is a third z dimension of the cuboid not shown, and coming out of the paper. Hence, this cuboid, unlike that shown in the figure, has six faces.
Accounting for the fluxes going through all the six faces, assigning the appropriate signs in accordance with the fluxes leaving and entering the cuboid, one arrives at
D · dS ≈ −Dx(x0, y0, z0)∆y∆z + Dx(x0 + ∆x, y0, z0)∆y∆z ∆S −Dy(x0, y0, z0)∆x∆z + Dy(x0, y0 + ∆y, z0)∆x∆z
−Dz(x0, y0, z0)∆x∆y + Dz(x0, y0, z0 + ∆z)∆x∆y (2.1.8)
Factoring out the volume of the cuboid ∆V = ∆x∆y∆z in the above, one gets
D · dS ≈ ∆V {[Dx(x0 + ∆x, . . .) − Dx(x0,...)] /∆x ∆S + [Dy(. . . , y0 + ∆y, . . .) − Dy(. . . , y0,...)] /∆y
+ [Dz(. . . , z0 + ∆z) − Dz(. . . , z0)] /∆z} (2.1.9)
Or that
D · dS ∂D ∂D ∂D ≈ x + y + z (2.1.10) ∆V ∂x ∂y ∂z In the limit when ∆V → 0, then
D · dS ∂D ∂D ∂D lim = x + y + z = ∇ · D (2.1.11) ∆V →0 ∆V ∂x ∂y ∂z Maxwell’s Equations, Differential Operator Form 21 where ∂ ∂ ∂ ∇ =x ˆ +y ˆ +z ˆ (2.1.12) ∂x ∂y ∂z
D =xD ˆ x +yD ˆ y +zD ˆ z (2.1.13) The above is the definition of the divergence operator in Cartesian coordinates. The diver- gence operator ∇· has its complicated representations in cylindrical and spherical coordinates, a subject that we would not delve into in this course. But they can be derived, and are best looked up at the back of some textbooks on electromagnetics. Consequently, one gets Gauss’s divergence theorem given by ¦ dV ∇ · D = D · dS (2.1.14) V S
2.1.2 Gauss’s Law in Differential Operator Form By further using Gauss’s or Coulomb’s law implies that ¦ D · dS = Q = dV % (2.1.15) S We can esplace the left-hand side of the above by (2.1.14) to arrive at ¦ ¦ dV ∇ · D = dV % (2.1.16) V V When V → 0, we arrive at the pointwise relationship, a relationship at an arbitrary point in space. Therefore,
∇ · D = % (2.1.17)
2.1.3 Physical Meaning of Divergence Operator The physical meaning of divergence is that if ∇·D 6= 0 at a point in space, it implies that there are fluxes oozing or exuding from that point in space [35]. On the other hand, if ∇ · D = 0, it implies no flux oozing out from that point in space. In other words, whatever flux that goes into the point must come out of it. The flux is termed divergence free. Thus, ∇ · D is a measure of how much sources or sinks exist for the flux at a point. The sum of these sources or sinks gives the amount of flux leaving or entering the surface that surrounds the sources or sinks. Moreover, if one were to integrate a divergence-free flux over a volume V , and invoking Gauss’s divergence theorem, one gets D · dS = 0 (2.1.18) S In such a scenerio, whatever flux that enters the surface S must leave it. In other words, what comes in must go out of the volume V , or that flux is conserved. This is true of incompressible 22 Electromagnetic Field Theory
fluid flow, electric flux flow in a source free region, as well as magnetic flux flow, where the flux is conserved.
Figure 2.4: In an incompressible flux flow, flux is conserved: whatever flux that enters a volume V must leave the volume V .
2.2 Stokes’s Theorem
The mathematical description of fluid flow was well established before the establishment of electromagnetic theory [36]. Hence, much mathematical description of electromagnetic theory uses the language of fluid. In mathematical notations, Stokes’s theorem is ¤ E · dl = ∇ × E · dS (2.2.1) C S
In the above, the contour C is a closed contour, whereas the surface S is not closed.3 First, applying Stokes’s theorem to a small surface ∆S, we define a curl operator4 ∇× at a point to be measured as
E · dl (∇ × E) · nˆ = lim ∆C (2.2.2) ∆S→0 ∆S
In the above, ∇ × E is a vector. Taking ∆C E · dl as a measure of the rotation of the field E around a small loop ∆C, the ratio of this rotation to the area of the loop ∆S has a limit when ∆S becomes infinitesimally small. This ratio is related to ∇ × E.
3In other words, C has no boundary whereas S has boundary. A closed surface S has no boundary like when we were proving Gauss’s divergence theorem previously. 4Sometimes called a rotation operator. Maxwell’s Equations, Differential Operator Form 23
Figure 2.5: In proving Stokes’s theorem, a closed contour C is assumed to enclose an open surface S. Then the surface S is tessellated into sum of small rects as shown. Stair-casing error vanishes in the limit when the rects are made vanishingly small.
First, the surface S enclosed by C is tessellated (also called meshed, gridded, or discretized) into sum of small rects (rectangles) as shown in Figure 2.5. Stokes’s theorem is then applied to one of these small rects to arrive at
Ei · dli = (∇ × Ei) · ∆Si (2.2.3) ∆Ci
where one defines ∆Si =n ˆ∆S. Next, we sum the above equation over i or over all the small rects to arrive at
X X Ei · dli = ∇ × Ei · ∆Si (2.2.4) i ∆Ci i
Again, on the left-hand side of the above, all the contour integrals over the small rects cancel each other internal to S save for those on the boundary. In the limit when ∆Si → 0, the left-hand side becomes a contour integral over the larger contour C, and the right-hand side becomes a surface integral over S. One arrives at Stokes’s theorem, which is
¤ E · dl = (∇ × E) · dS (2.2.5) C S 24 Electromagnetic Field Theory
Figure 2.6: We approximate the integration over a small rect using this figure. There are four edges to this small rect.
Next, we need to prove the details of definition (2.2.2) using Figure 2.6. Performing the integral over the small rect, one gets
E · dl = Ex(x0, y0, z0)∆x + Ey(x0 + ∆x, y0, z0)∆y ∆C − Ex(x0, y0 + ∆y, z0)∆x − Ey(x0, y0, z0)∆y E (x , y , z ) E (x , y + ∆y, z ) = ∆x∆y x 0 0 0 − x 0 0 0 ∆y ∆y E (x , y , z ) E (x , y + ∆y, z ) − y 0 0 0 + y 0 0 0 ∆x ∆x (2.2.6)
We have picked the normal to the incremental surface ∆S to bez ˆ in the above example, and hence, the above gives rise to the identity that
∆S E · dl ∂ ∂ lim = Ey − Ex =z ˆ · ∇ × E (2.2.7) ∆S→0 ∆S ∂x ∂y Picking different ∆S with different orientations and normalsn ˆ where |hatn =x ˆ orn ˆ =y ˆ, one gets ∂ ∂ E − E =x ˆ · ∇ × E (2.2.8) ∂y z ∂z y ∂ ∂ E − E =y ˆ · ∇ × E (2.2.9) ∂z x ∂x z Maxwell’s Equations, Differential Operator Form 25
The above gives the x, y, and z components of ∇ × E. It is to be noted that ∇ × E is a vector. In other words, one gets
∂ ∂ ∂ ∂ ∇ × E =x ˆ E − E +y ˆ E − E ∂y z ∂z y ∂z x ∂x z ∂ ∂ +ˆz E − E (2.2.10) ∂x y ∂y x where ∂ ∂ ∂ ∇ =x ˆ +y ˆ +z ˆ (2.2.11) ∂x ∂y ∂z
2.2.1 Faraday’s Law in Differential Operator Form Faraday’s law is experimentally motivated. Michael Faraday (1791-1867) was an extraordi- nary experimentalist who documented this law with meticulous care. It was only decades later that a mathematical description of this law was arrived at. Faraday’s law in integral form is given by5 ¤ d E · dl = − B · dS (2.2.12) C dt S Assuming that the surface S is not time varying, one can take the time derivative into the integrand and write the above as ¤ ∂ E · dl = − B · dS (2.2.13) C S ∂t One can replace the left-hand side with the use of Stokes’ theorem to arrive at ¤ ¤ ∂ ∇ × E · dS = − B · dS (2.2.14) S S ∂t The normal of the surface element dS can be pointing in an arbitrary direction, and the surface S can be made very small. Then the integral can be removed, and one has
∂ ∇ × E = − B (2.2.15) ∂t The above is Faraday’s law in differential operator form. ∂B In the static limit, ∂t = 0, giving
∇ × E = 0 (2.2.16)
5Faraday’s law is experimentally motivated. Michael Faraday (1791-1867) was an extraordinary exper- imentalist who documented this law with meticulous care. It was only decades later that a mathematical description of this law was arrived at. 26 Electromagnetic Field Theory
2.2.2 Physical Meaning of Curl Operator
The curl operator ∇× is a measure of the rotation or the circulation of a field at a point in space. On the other hand, ∆C E · dl is a measure of the circulation of the field E around the loop formed by C. Again, the curl operator has its complicated representations in other co- ordinate systems like cylindrical or spherical coordinates, a subject that will not be discussed in detail here. It is to be noted that our proof of the Stokes’s theorem is for a flat open surface S, and not for a general curved open surface. Since all curved surfaces can be tessellated into a union of flat triangular surfaces according to the tiling theorem, the generalization of the above proof to curved surface is straightforward. An example of such a triangulation of a curved surface into a union of flat triangular surfaces is shown in Figure 2.7.
Figure 2.7: An arbitrary curved surface can be triangulated with flat triangular patches. The triangulation can be made arbitrarily accurate by making the patches arbitrarily small.
2.3 Maxwell’s Equations in Differential Operator Form
With the use of Gauss’ divergence theorem and Stokes’ theorem, Maxwell’s equations can be written more elegantly in differential operator forms. They are:
∂B ∇ × E = − (2.3.1) ∂t ∂D ∇ × H = + J (2.3.2) ∂t ∇ · D = % (2.3.3) ∇ · B = 0 (2.3.4)
These equations are point-wise relations as they relate the left-hand side and right-hand side field values at a given point in space. Moreover, they are not independent of each other. For instance, one can take the divergence of the first equation (2.3.1), making use of the vector Maxwell’s Equations, Differential Operator Form 27 identity that ∇ · (∇ × E) = 0, one gets
∂∇ · B − = 0 → ∇ · B = constant (2.3.5) ∂t This constant corresponds to magnetic charges, and since they have not been experimentally observed, one can set the constant to zero. Thus the fourth of Maxwell’s equations, (2.3.4), follows from the first (2.3.1). Similarly, by taking the divergence of the second equation (2.3.2), and making use of the current continuity equation that
∂% ∇ · J + = 0 (2.3.6) ∂t one can obtain the second last equation (2.3.3). Notice that in (2.3.3), the charge density % can be time-varying, whereas in the previous lecture, we have “derived” this equation from Coulomb’s law using electrostatic theory. The above logic follows if ∂/∂t 6= 0, and is not valid for static case. In other words, for statics, the third and the fourth equations are not derivable from the first two. Hence all four Maxwell’s equations are needed for static problems. For electrodynamic problems, only solving the first two suffices. Something is amiss in the above. If J is known, then solving the first two equations implies solving for four vector unknowns, E, H, B, D, which has 12 scalar unknowns. But there are only two vector equations or 6 scalar equations in the first two equations. Thus one needs more equations. These are provided by the constitutive relations that we shall discuss next.
2.4 Homework Examples
Example 1 If D = (2y2 + z)ˆx + 4xyyˆ + xzˆ, find:
1. Volume charge density ρv at (−1, 0, 3).
2. Electric flux through the cube defined by
0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.
3. Total charge enclosed by the cube.
Example 2 ¢ Suppose E = ˆx3y + ˆyx, calculate E · dl along a straight line in the x-y plane joining (0,0) to (3,1). 28 Electromagnetic Field Theory 2.5 Historical Notes
There are several interesting historical notes about Maxwell.
It is to be noted that when James Clerk Maxwell first wrote his equations down, it was in many equations and very difficult to digest [17, 37, 38]. It was Oliver Heaviside who distilled those equations and presented them in the present form found in textbooks. Putatively, most cannot read his treatise [37] beyond the first 50 pages [39]. Maxwell wrote many poems in his short lifespan (1831-1879) and they can be found at [40]. Also, the ancestor of James Clerk Maxwell married from the Clerk family into the Maxwell family. One of the conditions of marriage was that all the descendants of the Clerk family would be named Clerk Maxwell. That was why Maxwell is addressed as Professor Clerk Maxwell. Lecture 3
Constitutive Relations, Wave Equation, Electrostatics, and Static Green’s Function
Constitutive relations are important for defining the electromagnetic material properties of the media involved. Also, wave phenomenon is a major triumph of Maxwell’s equations. Hence, we will study that derivation of this phenomenon here. To make matter simple, we will reduce the problem to electrostatics to simplify the math to introduce the concept of the Green’s function. As mentioned previously, for time-varying problems, only the first two of the four Maxwell’s equations suffice. The latter two are derivable from the first two. But the first two equations have four unknowns E, H, D, and B. Hence, two more equations are needed to solve for these unknowns. These equations come from the constitutive relations.
3.1 Simple Constitutive Relations
The constitution relation between electric flux D and the electric field E in free space (or vacuum) is
D = ε0E (3.1.1)
When material medium is present, one has to add the contribution to D by the polarization density P which is a dipole density.1 Then [31, 32,41]
D = ε0E + P (3.1.2)
1Note that a dipole moment is given by Q` where Q is its charge in coulomb and ` is its length in m. Hence, dipole density, or polarization density as dimension of coulomb/m2, which is the same as that of electric flux D.
29 30 Electromagnetic Field Theory
The second term P above is due to material property, and the contribution to the electric flux due to the polarization density of the medium. It is due to the little dipole contribution due to the polar nature of the atoms or molecules that make up a medium. By the same token, the first term ε0E can be thought of as the polarization density contribution of vacuum. Vacuum, though represents nothingness, has electrons and positrons, or electron-positron pairs lurking in it [42]. Electron is matter, whereas positron is anti- matter. In the quiescent state, they represent nothingness, but they can be polarized by an electric field E. That also explains why electromagnetic wave can propagate through vacuum. For many media, they are approximately linear media. Then P is linearly proportional to E, or P = ε0χ0E, or
D = ε0E + ε0χ0E
= ε0(1 + χ0)E = εE, ε = ε0(1 + χ0) = ε0εr (3.1.3) where χ0 is the electric susceptibility . In other words, for linear material media, one can replace the vacuum permittivity ε0 with an effective permittivity ε = ε0εr where εr is the relative permittivity. Thus, D is linearly proportional to E. In free space,2
10−8 ε = ε = 8.854 × 10−12 ≈ F/m (3.1.4) 0 36π The constitutive relation between magnetic flux B and magnetic field H is given as
B = µH, µ = permeability H/m (3.1.5)
In free space or vacuum,
−7 µ = µ0 = 4π × 10 H/m (3.1.6)
As shall be explained later, this is an assigned value giving it a precise value as shown above. In other materials, the permeability can be written as
µ = µ0µr (3.1.7)
The above can be derived using similar argument for permittivity, where the different perme- ability is due to the presence of magnetic dipole density in a material medium. In the above, µr is termed the relative permeability.
3.2 Emergence of Wave Phenomenon, Triumph of Maxwell’s Equations
One of the major triumphs of Maxwell’s equations is the prediction of the wave phenomenon. This was experimentally verified by Heinrich Hertz in 1888 [18], some 23 years after the
2It is to be noted that we will use MKS unit in this course. Another possible unit is the CGS unit used in many physics texts [43] Constitutive Relations, Wave Equation, Electrostatics, and Static Green’s Function 31 completion of Maxwell’s theory in 1865 [17]. To see this, we consider the first two Maxwell’s equations for time-varying fields in vacuum or a source-free medium.3 They are ∂H ∇ × E = −µ (3.2.1) 0 ∂t ∂E ∇ × H = −ε (3.2.2) 0 ∂t Taking the curl of (3.2.1), we have
∂ ∇ × ∇ × E = −µ ∇ × H (3.2.3) 0 ∂t It is understood that in the above, the double curl operator implies ∇×(∇×E). Substituting (3.2.2) into (3.2.3), we have
∂2 ∇ × ∇ × E = −µ ε E (3.2.4) 0 0 ∂t2 In the above, the left-hand side can be simplified by using the identity that a × (b × c) = b(a · c) − c(a · b),4 but be mindful that the operator ∇ has to operate on a function to its right. Therefore, we arrive at the identity that
∇ × ∇ × E = ∇∇ · E − ∇2E (3.2.5)
Since ∇ · E = 0 in a source-free medium, we have
∂2 ∇2E − µ ε E = 0 (3.2.6) 0 0 ∂t2 where ∂2 ∂2 ∂2 ∇2 = ∇ · ∇ = + + ∂x2 ∂y2 ∂z2 The above is known as the Laplacian operator. Here, (3.2.6) is the wave equation in three space dimensions [32,44]. To see the simplest form of wave emerging in the above, we can let E =xE ˆ x(z, t) so that ∇ · E = 0 satisfying the source-free condition. Then (3.2.6) becomes
∂2 ∂2 E (z, t) − µ ε E (z, t) = 0 (3.2.7) ∂z2 x 0 0 ∂t2 x Eq. (3.2.7) is known mathematically as the wave equation in one space dimension. It can also be written as ∂2 1 ∂2 2 f(z, t) − 2 2 f(z, t) = 0 (3.2.8) ∂z c0 ∂t
3 Since the third and the fourth Maxwell’s equations are derivable from the first two when ∂/∂t 6= 0. 4For mnemonics, this formula is also known as the “back-of-the-cab” formula. 32 Electromagnetic Field Theory
2 −1 where c0 = (µ0ε0) . Eq. (3.2.8) can also be factorized as
∂ 1 ∂ ∂ 1 ∂ − + f(z, t) = 0 (3.2.9) ∂z c0 ∂t ∂z c0 ∂t or
∂ 1 ∂ ∂ 1 ∂ + − f(z, t) = 0 (3.2.10) ∂z c0 ∂t ∂z c0 ∂t
The above can be verified easily by direct expansion, and using the fact that
∂ ∂ ∂ ∂ = (3.2.11) ∂t ∂z ∂z ∂t
The above implies that we have either
∂ 1 ∂ + f+(z, t) = 0 (3.2.12) ∂z c0 ∂t or
∂ 1 ∂ − f−(z, t) = 0 (3.2.13) ∂z c0 ∂t
Equation (3.2.12) and (3.2.13) are known as the one-way wave equations or the advective equations [45]. From the above factorization, it is seen that the solutions of these one-way wave equations are also the solutions of the original wave equation given by (3.2.8). Their general solutions are then
f+(z, t) = F+(z − c0t) (3.2.14)
f−(z, t) = F−(z + c0t) (3.2.15)
We can verify the above by back substitution into (3.2.12) and (3.2.13). Eq. (3.2.14) consti- tutes a right-traveling wave function of any shape while (3.2.15) constitutes a left-traveling wave function of any shape. Since Eqs. (3.2.14) and (3.2.15) are also solutions to (3.2.8), we can write the general solution to the wave equation as
f(z, t) = F+(z − c0t) + F−(z + c0t) (3.2.16)
This is a wonderful result since F+ and F− are arbitrary functions of any shape (see Figure 3.1); they can be used to encode information for communication! Constitutive Relations, Wave Equation, Electrostatics, and Static Green’s Function 33
Figure 3.1: Solutions of the wave equation can be a single-valued function of any shape. In the above, F+ travels in the positive z direction, while F− travels in the negative z direction as t increases.
Furthermore, one can calculate the velocity of this wave to be
8 c0 = 299, 792, 458m/s ' 3 × 10 m/s (3.2.17) p where c0 = 1/µ0ε0. Maxwell’s equations (3.2.1) implies that E and H are linearly proportional to each other. Thus, there is only one independent constant in the wave equation, and the value of µ0 is −7 −1 defined neatly to be 4π × 10 henry m , while the value of ε0 has been measured to be about 8.854 × 10−12 farad m−1. Now it has been decided that the velocity of light is used as a standard and is defined to be the integer given in (3.2.17). A meter is defined to be the distance traveled by light in 1/(299792458) seconds. Hence, the more accurate that unit of time or second can be calibrated, the more accurate can we calibrate the unit of length or meter. Therefore, the design of an accurate clock like an atomic clock is an important research problem. The value of ε0 was measured in the laboratory quite early. Then it was realized that electromagnetic wave propagates at a tremendous velocity which is the velocity of light.5 This
5The velocity of light was known in astronomy by (Roemer, 1676) [19]. 34 Electromagnetic Field Theory was also the defining moment which revealed that the field of electricity and magnetism and the field of optics were both described by Maxwell’s equations or electromagnetic theory.
3.3 Static Electromagnetics–Revisted
We have seen static electromagnetics previously in integral form. Now we look at them in differential operator form. When the fields and sources are not time varying, namely that ∂/∂t = 0, we arrive at the static Maxwell’s equations for electrostatics and magnetostatics, namely [31, 32,46]
∇ × E = 0 (3.3.1) ∇ × H = J (3.3.2) ∇ · D = % (3.3.3) ∇ · B = 0 (3.3.4)
Notice the the electrostatic field system is decoupled from the magnetostatic field system. However, in a resistive system where
J = σE (3.3.5) the two systems are coupled again. This is known as resistive coupling between them. But if σ → ∞, in the case of a perfect conductor, or superconductor, then for a finite J, E has to be zero. The two systems are decoupled again. Also, one can arrive at the equations above by letting µ0 → 0 and 0 → 0. In this case, the velocity of light becomes infinite, or retardation effect is negligible. In other words, there is no time delay for signal propagation through the system in the static approximation. Finally, it is important to note that in statics, the latter two Maxwell’s equations are not derivable from the first two. Hence, all four equations have to be considered when one seeks the solution in the static regime or the long-wavelength regime.
3.3.1 Electrostatics We see that Faraday’s law in the static limit is
∇ × E = 0 (3.3.6)
One way to satisfy the above is to let E = −∇Φ because of the identity ∇ × ∇ = 0.6 Alternatively, one can assume that E is a constant. But we usually are interested in solutions that vanish at infinity, and hence, the latter is not a viable solution. Therefore, we let
E = −∇Φ (3.3.7)
6One an easily go through the algebra in cartesian coordinates to convince oneself of this. Constitutive Relations, Wave Equation, Electrostatics, and Static Green’s Function 35
3.3.2 Poisson’s Equation As a consequence of the above,
∇ · D = % ⇒ ∇ · εE = % ⇒ −∇ · ε∇Φ = % (3.3.8)
In the last equation above, if ε is a constant of space, or independent of r, then one arrives at the simple Poisson’s equation, which is a partial differential equation % ∇2Φ = − (3.3.9) ε Here, the Laplacian operator
∂2 ∂2 ∂2 ∇2 = ∇ · ∇ = + + ∂x2 ∂y2 ∂z2
For a point source, we know from Coulomb’s law that q E = rˆ = −∇Φ (3.3.10) 4πεr2 From the above, we deduce that7 q Φ = (3.3.11) 4πεr Therefore, we know the solution to Poisson’s equation (3.3.9) when the source % represents a point source. Since this is a linear equation, we can use the principle of linear superposition to find the solution when the charge density %(r) is arbitrary. A point source located at r0 is described by a charge density as
%(r) = qδ(r − r0) (3.3.12) where δ(r−r0) is a short-hand notation for δ(x−x0)δ(y −y0)δ(z −z0). Therefore, from (3.3.9), the corresponding partial differential equation for a point source is
qδ(r − r0) ∇2Φ(r) = − (3.3.13) ε The solution to the above equation, from Coulomb’s law in accordance to (3.3.11), has to be q Φ(r) = (3.3.14) 4πε|r − r0| whereas (3.3.11) is for a point source at the origin, (3.3.14) is for a point source located and translated to r0. The above is a coordinate independent form of the solution. Here, r =xx ˆ +yy ˆ +zz ˆ and r0 =xx ˆ 0 +yy ˆ 0 +zz ˆ 0, and |r − r0| = p(x − x0)2 + (y − y0)2 + (z − z0)2.
7One can always take the gradient or ∇ of Φ to verify this. Mind you, this is best done in spherical coordinates. 36 Electromagnetic Field Theory
3.3.3 Static Green’s Function Let us define a partial differential equation given by ∇2g(r − r0) = −δ(r − r0) (3.3.15) The above is similar to Poisson’s equation with a point source on the right-hand side as in (3.3.13). Such a solution, a response to a point source, is called the Green’s function.8 By comparing equations (3.3.13) and (3.3.15), then making use of (3.3.14), we deduced that the static Green’s function is 1 g(r − r0) = (3.3.16) 4π|r − r0| An arbitrary source can be expressed as ¦ %(r) = dV 0%(r0)δ(r − r0) (3.3.17) V The above is just the statement that an arbitrary charge distribution %(r) can be expressed as a linear superposition of point sources δ(r − r0). Using the above in (3.3.9), we have ¦ 1 ∇2Φ(r) = − dV 0%(r0)δ(r − r0) (3.3.18) ε V We can let ¦ 1 Φ(r) = dV 0g(r − r0)%(r0) (3.3.19) ε V By substituting the above into the left-hand side of (3.3.18), exchanging order of integration and differentiation, and then making use of equation (3.3.15), it can be shown that (3.3.19) indeed satisfies (3.3.9). The above is just a convolutional integral. Hence, the potential Φ(r) due to an arbitrary source distribution %(r) can be found by using convolution, namely, ¦ 0 1 %(r ) 0 Φ(r) = 0 dV (3.3.20) 4πε V |r − r | In a nutshell, the solution of Poisson’s equation when it is driven by an arbitrary source %, is the convolution of the source with the static Green’s function, a point source response.
3.3.4 Laplace’s Equation If % = 0, or if we are in a source-free region, then ∇2Φ =0 (3.3.21) which is the Laplace’s equation. Laplace’s equation is usually solved as a boundary value problem. In such a problem, the potential Φ is stipulated on the boundary of a region with a certain boundary condition, and then the solution is sought in the region so as to match the boundary condition. Examples of such boundary value problems are given at the end of the lecture.
8George Green (1793-1841), the son of a Nottingham miller, was self-taught, but his work has a profound impact in our world. Constitutive Relations, Wave Equation, Electrostatics, and Static Green’s Function 37
3.4 Homework Examples
Example 1
Fields of a sphere of radius a with uniform charge density ρ:
Figure 3.2: Figure of a sphere with uniform charge density for the example above.
Assuming that Φ|r=∞ = 0, what is Φ at r ≤ a? And Φ at r > a.
Example 2
A capacitor has two parallel plates attached to a battery, what is E field inside the capacitor? First, one guess the electric field between the two parallel plates. Then one arrive at a potential Φ in between the plates so as to produce the field. Then the potential is found so as to match the boundary conditions of Φ = V in the upper plate, and Φ = 0 in the lower plate. What is the Φ that will satisfy the requisite boundary condition?
Figure 3.3: Figure of a parallel plate capacitor. The field in between can be found by solving Laplace’s equation as a boundary value problem [31].
Example 3
A coaxial cable has two conductors. The outer conductor is grounded and hence is at zero potential. The inner conductor is at voltage V . What is the solution? 38 Electromagnetic Field Theory
For this, one will have to write the Laplace’s equation in cylindrical coordinates, namely,
1 ∂ ∂Φ 1 ∂2Φ ∇2Φ = ρ + = 0 (3.4.1) ρ ∂ρ ∂ρ ρ2 ∂φ2
In the above, we assume that the potential is constant in the z direction, and hence, ∂/∂z = 0, and ρ, φ, z are the cylindrical coordinates. By assuming axi-symmetry, we can let ∂/∂φ = 0 and the above becomes 1 ∂ ∂Φ ∇2Φ = ρ = 0 (3.4.2) ρ ∂ρ ∂ρ
Show that Φ = A ln ρ + B is a general solution to Laplace’s equation in cylindrical coor- dinates inside a coax. What is the Φ that will satisfy the requisite boundary condition?
Figure 3.4: The field in between a coaxial line can also be obtained by solving Laplace’s equation as a boundary value problem (courtesy of Ramo, Whinnery, and Van Duzer [31]). Lecture 4
Magnetostatics, Boundary Conditions, and Jump Conditions
In the previous lecture, Maxwell’s equations become greatly simplified in the static limit. We have looked at how the electrostatic problems are solved. We now look at the magnetostatic case. In addition, we will study boundary conditions and jump conditions at an interface, and how they are derived from Maxwell’s equations. Maxwell’s equations can be first solved in different domains. Then the solutions are pieced (or sewn) together by imposing boundary conditions at the boundaries or interfaces of the domains. Such problems are called boundary- value problems (BVPs).
4.1 Magnetostatics
From Maxwell’s equations, we can deduce that the magnetostatic equations for the magnetic field and flux when ∂/∂t = 0, which are [31, 32,46]
∇ × H = J (4.1.1) ∇ · B = 0 (4.1.2)
These two equations are greatly simplified, and hence, are easier to solve compared to the time-varying case. One way to satisfy the second equation is to let
B = ∇ × A (4.1.3) because of the vector identity
∇ · (∇ × A) = 0 (4.1.4)
39 40 Electromagnetic Field Theory
The above is zero for the same reason that a · (a × b) = 0. In this manner, Gauss’s law in (4.1.2) is automatically satisfied. From (4.1.1), we have
B ∇ × = J (4.1.5) µ
Then using (4.1.3) into the above,
1 ∇ × ∇ × A = J (4.1.6) µ
In a homogeneous medium,1 µ or 1/µ is a constant and it commutes with the differential ∇ operator or that it can be taken outside the differential operator. As such, one arrives at
∇ × (∇ × A) = µJ (4.1.7)
We use the vector identity that (see back-of-cab formula in the previous lecture)
∇ × (∇ × A) = ∇(∇ · A) − (∇ · ∇)A = ∇(∇ · A) − ∇2A (4.1.8) where ∇2 is a shorthand notation for ∇ · ∇. As a result, we arrive at [47]
∇(∇ · A) − ∇2A = µJ (4.1.9)
By imposing the Coulomb gauge that ∇ · A = 0, which will be elaborated in the next section, we arrive at the simplified equation
∇2A = −µJ (4.1.10)
The above is also known as the vector Poisson’s equation. In cartesian coordinates, the above can be viewed as three scalar Poisson’s equations. Each of the Poisson’s equation can be solved using the Green’s function method previously described. Consequently, in free space ¦ µ J(r0) A(r) = dV 0 (4.1.11) 4π V R where
R = |r − r0| (4.1.12) is the distance between the source point r0 and the observation point r. Here, dV 0 = dx0dy0dz0. It is also variously written as dr0 or d3r0.
1Its prudent to warn the reader of the use of the word “homogeneous”. In the math community, it usually refers to something to be set to zero. But in the electromagnetics community, it refers to something “non- heterogeneous”. Magnetostatics, Boundary Conditions, and Jump Conditions 41
4.1.1 More on Coulomb Gauge Gauge is a very important concept in physics [48], and we will further elaborate it here. First, notice that A in (4.1.3) is not unique because one can always define
A0 = A − ∇Ψ (4.1.13)
Then
∇ × A0 = ∇ × (A − ∇Ψ) = ∇ × A = B (4.1.14) where we have made use of that ∇ × ∇Ψ = 0. Hence, the ∇× of both A and A0 produce the same B; hence, A is non-unique. To find A uniquely, we have to define or set the divergence of A or provide a gauge condition. One way is to set the divergence of A to be zero, namely that
∇ · A = 0 (4.1.15)
Then
∇ · A0 = ∇ · A − ∇2Ψ 6= ∇ · A (4.1.16)
The last non-equal sign follows if ∇2Ψ 6= 0. However, if we further stipulate that ∇ · A0 = ∇ · A = 0, then −∇2Ψ = 0. This does not necessary imply that Ψ = 0, but if we impose that condition that Ψ → 0 when r → ∞, then Ψ = 0 everywhere.2 By so doing, A and A0 are equal to each other, and we obtain (4.1.10) and (4.1.11). The above is akin to the idea that given a vector a, just by stipulating that b × a = c is not enough to determine a. We need to stipulate what b · a is as well. Here, a, b, and c are independent vectors.
4.2 Boundary Conditions–1D Poisson’s Equation
To simplify the solutions of Maxwell’s equations, they are usually solved in a homogeneous medium. As mentioned before, a complex problem can be divided into piecewise homogeneous regions first, and then the solution in each region sought. Then the total solution must satisfy boundary conditions at the interface between the piecewise homogeneous regions. What are these boundary conditions? Boundary conditions are actually embedded in the partial differential equations that the potential or the field satisfy. Two important concepts to keep in mind are:
Differentiation of a function with discontinuous slope will give rise to step discontinuity. Differentiation of a function with step discontinuity will give rise to a Dirac delta func- tion. This is also called the jump condition, a term often used by the mathematics community [49].
2It is a property of the Laplace boundary value problem that if Ψ = 0 on a closed surface S, then Ψ = 0 everywhere inside S. Earnshaw’s theorem [31] is useful for proving this assertion. 42 Electromagnetic Field Theory
Take for example a one dimensional Poisson’s equation that d d ε(x) Φ(x) = −%(x) (4.2.1) dx dx where ε(x) represents material property that has the form given in Figure 4.1. One can actually say something about Φ(x) given %(x) on the right-hand side. If %(x) has a delta d function singularity, it implies that ε(x) dx Φ(x) has a step discontinuity. If %(x) is finite d everywhere, then ε(x) dx Φ(x) must be continuous everywhere. d Furthermore, if ε(x) dx Φ(x) is finite everywhere, it implies that Φ(x) must be continuous everywhere.
Figure 4.1: A figure showing a charge sheet at the interface between two dielectric media. Because it is a surface charge sheet, the volume charge density %(x) is infinite at the sheet location x0.
To see this in greater detail, we illustrate it with the following example. In the above, %(x) represents a charge distribution given by %(x) = %sδ(x − x0). In this case, the charge distribution is everywhere zero except at the location of the surface charge sheet, where the charge density is infinite: it is represented mathematically by a delta function3 in space. To find the boundary condition of the potential Φ(x) at x0 , we integrate (4.2.1) over an infinitesimal width around x0, the location of the charge sheet, namely ¢ ¢ ¢ x0+∆ d d x0+∆ x0+∆ dx ε(x) Φ(x) = − dx%(x) = − dx%0δ(x − x0) (4.2.2) x0−∆ dx dx x0−∆ x0−∆ Since the integrand of the left-hand side is an exact derivative, we get
x0+∆ d ε(x) Φ(x) = −%s (4.2.3) dx x0−∆ 3This function has been attributed to Dirac who used in pervasively, but Cauchy was aware of such a function. Magnetostatics, Boundary Conditions, and Jump Conditions 43 whereas on the right-hand side, we pick up the contribution from the delta function. Evalu- ating the left-hand side at their limits, one arrives at
d d ε(x+) Φ(x+) − ε(x−) Φ(x−) =∼ −% , (4.2.4) 0 dx 0 0 dx 0 s
± d where x0 = lim∆→0 x0 ± ∆. In other words, the jump discontinuity is in ε(x) dx Φ(x) and the amplitude of the jump discontinuity is proportional to the amplitude of the delta function. Since E = −∇Φ, or that
d E (x) = − Φ(x), (4.2.5) x dx
The above implies that
+ + − − ε(x0 )Ex(x0 ) − ε(x0 )Ex(x0 ) = %s (4.2.6) or
+ − Dx(x0 ) − Dx(x0 ) = %s (4.2.7) where
Dx(x) = ε(x)Ex(x) (4.2.8)
The lesson learned from above is that boundary condition is obtained by integrating the pertinent differential equation over an infinitesimal small segment. In this mathematical way of looking at the boundary condition, one can also eyeball the differential equation and ascertain the terms that will have the jump discontinuity whose derivatives will yield the delta function on the right-hand side.
4.3 Boundary Conditions–Maxwell’s Equations
As seen previously, boundary conditions for a field is embedded in the differential equation that the field satisfies. Hence, boundary conditions can be derived from the differential operator forms of Maxwell’s equations. In most textbooks, boundary conditions are obtained by integrating Maxwell’s equations over a small pill box [31,32,47]. To derive these boundary conditions, we will take an unconventional view: namely to see what sources can induce jump conditions on the pertinent fields. Boundary conditions are needed at media interfaces, as well as across current or charge sheets. As shall be shown, each of the Maxwell’s equations induces a boundary condition at the interface between two media or two regions separated by surface sources. 44 Electromagnetic Field Theory
4.3.1 Faraday’s Law
Figure 4.2: This figure is for the derivation of boundary condition induced by Faraday’s law. A local coordinate system can be used to see the boundary condition more lucidly. Here, the normaln ˆ =y ˆ and the tangential component tˆ=x ˆ.
For this, we start with Faraday’s law, which implies that
∂B ∇ × E = − (4.3.1) ∂t The right-hand side of this equation is a derivative of a time-varying magnetic flux, and it is a finite quantity. One quick answer we could ask is that if the right-hand side of the above equation is everywhere finite, could there be any jump discontinuity on the field E on the left hand side? The answer is no. To see this quickly, one can project the tangential field component and normal field component to a local coordinate system. In other words, one can think of tˆ andn ˆ as the localx ˆ andy ˆ coordinates. Then writing the curl operator in this local coordinates, one gets
∂ ∂ ∇ × E = xˆ +y ˆ × (ˆxE +yE ˆ ) ∂x ∂y x y ∂ ∂ =z ˆ E − zˆ E (4.3.2) ∂x y ∂y x
In simplifying the above, we have used the distributive property of cross product, and eval- uating the cross product in cartesian coordinates. The cross product gives four terms, but only two of the four terms are non-zero as shown above. ∂ ∂ Since the right-hand side of (4.3.1) is finite, the above implies that ∂x Ey and ∂y Ex have to be finite. In order words, Ex is continuous in the y direction and Ey is continuous in the x direction. Since in the local coordinate system, Ex = Et, then Et is continuous across the boundary. The above implies that
E1t = E2t (4.3.3) Magnetostatics, Boundary Conditions, and Jump Conditions 45 or the tangential components of the electric field is continuous at the interface. To express this in a coordinate independent manner, we have
nˆ × E1 =n ˆ × E2 (4.3.4) wheren ˆ is the unit normal at the interface, andn ˆ × E always bring out the tangential component of a vector E (convince yourself).
4.3.2 Gauss’s Law for Electric Flux From this Gauss’s law, we have
∇ · D = % (4.3.5) where % is the volume charge density.
Figure 4.3: A figure showing the derivation of boundary condition for Gauss’s law. Again, a local coordinate system can be introduced for simplicity.
Expressing the above in local coordinates (x, y, z as shown in Figure 4.3, then
∂ ∂ ∂ ∇ · D = D + D + D = % (4.3.6) ∂x x ∂y y ∂z z The boundary condition for the electric flux can be found by singularity matching. If there is a surface layer charge at the interface, then the volume charge density must be infinitely large or singular; hence, it can be expressed in terms of a delta function, or % = %sδ(z) in local coordinates. By looking at the above expression, the only term that can produce a ∂ δ(z) is from ∂z Dz. In other words, Dz has a jump discontinuity at z = 0; the other terms do not. Then ∂ D = % δ(z) (4.3.7) ∂z z s 46 Electromagnetic Field Theory
Integrating the above from 0 − ∆ to 0 + ∆, we get
0+∆
Dz(z) = %s (4.3.8) 0−∆ or in the limit when ∆ → 0,
+ − Dz(0 ) − Dz(0 ) = %s (4.3.9)
+ − + − where 0 = lim∆→0 0 + ∆, and 0 = lim∆→0 0 − ∆. Since Dz(0 ) = D2n, Dz(0 ) = D1n, the above becomes
D2n − D1n = %s (4.3.10)
In other words, a charge sheet %s can give rise to a jump discontinuity in the normal component of the electric flux D. Expressed in coordinate independent form, it is
nˆ · (D2 − D1) = %s (4.3.11)
Using the physical notion that an electric charge has electric flux D exuding from it, Figure 4.4 shows an intuitive sketch as to why a charge sheet gives rise to a discontinuous normal component of the electric flux D.
Figure 4.4: A figure intuitively showing why a sheet of charge gives rise to a jump discontinuiy in the normal component of the electric flux D.
4.3.3 Ampere’s Law Ampere’s law, or the generalized one, stipulates that ∂D ∇ × H = J + (4.3.12) ∂t Again if the right-hand side is everywhere finite, then H is a continuous field everywhere. However, if the right-hand side has a delta function singularity, due to a current sheet in 3D ∂D space, and that ∂t is regular or finite everywhere, then the only place where the singularity can be matched on the left-hand side is from the derivative of the magnetic field H or ∇ × H. In a word, H is not continuous. For instance, we can project the above equation onto a local coordinates just as we did for Faraday’s law. Magnetostatics, Boundary Conditions, and Jump Conditions 47
Figure 4.5: A figure showing the derivation of boundary condition for Ampere’s law. A local coordinate system is used for simplicity.
To be general, we also include the presence of a current sheet at the interface. A current sheet, or a surface current density becomes a delta function singularity when expressed as a volume current density; Thus, rewriting (4.3.12) in a local coordinate system, assuming that 4 J =xJ ˆ sxδ(z), then singularity matching, ∂ ∂ ∇ × H =x ˆ H − H =xJ ˆ δ(z) (4.3.13) ∂y z ∂z y sx The displacement current term on the right-hand side is ignored since it is regular or finite, and will not induce a jump discontinuity on the field; hence, we have the form of the right- hand side of the above equation. From the above, the only term that can produce a δ(z) ∂ singularity on the left-hand side is the − ∂z Hy term. Therefore, by singularity matching, we conclude that ∂ − H = J δ(z) (4.3.14) ∂z y sx
In other words, Hy has to have a jump discontinuity at the interface where the current sheet resides. Or that
+ − Hy(z = 0 ) − Hy(z = 0 ) = −Jsx (4.3.15) The above implies that
H2y − H1y = −Jsx (4.3.16)
But Hy is just the tangential component of the H field. In a word, the current sheet Jsx induces a jump discontinuity on the y component of the magnetic field. Now if we repeat the same exercise with a current with a y component, or J =yJ ˆ syδ(z), at the interface, we have
H2x − H1x = Jsy (4.3.17)
4The form of this equation can be checked by dimensional analysis. Here, J has the unit of A m−2, δ(z) −1 −1 has unit of m , and Jsx, a current sheet density, has unit of A m . 48 Electromagnetic Field Theory
Now, (4.3.16) and (4.3.17) can be rewritten using a cross product as
zˆ × (ˆyH2y − yHˆ 1y) =xJ ˆ sx (4.3.18)
zˆ × (ˆxH2x − xHˆ 1x) =yJ ˆ sy (4.3.19)
The above two equations can be combined as one, written in a coordinate independent form, to give
nˆ × (H2 − H1) = Js (4.3.20) where in this case here,n ˆ =z ˆ. In other words, a current sheet Js can give rise to a jump discontinuity in the tangential components of the magnetic field,n ˆ × H. This is illustrated intuitively in Figure 4.6.
Figure 4.6: A figure intuitively showing that with the understanding of how a single line current source generates a magnetic field (right), a cluster of them forming a current sheet will generate a jump discontinuity in the tangential component of the magnetic field H (left).
4.3.4 Gauss’s Law for Magnetic Flux Similarly, from Gauss’s law for magnetic flux, or that
∇ · B = 0 (4.3.21) one deduces that
nˆ · (B2 − B1) = 0 (4.3.22) or that the normal magnetic fluxes are continuous at an interface. In other words, since mag- netic charges do not exist, the normal component of the magnetic flux has to be continuous. The take-home message here is that the boundary conditions are buried in the differential operators. If there singular terms in Maxwell’s equations, then via the differential operators, the boundary conditions can be deduced. These boundary conditions are also known as jump condition if a current or a source sheet is present. Lecture 5
Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem
Biot-Savart law, like Ampere’s law was experimentally determined in around 1820 and it is discussed in a number of textbooks [31, 32, 48]. This is the cumulative work of Ampere, Oersted, Biot, and Savart. At this stage of the course, one has the mathematical tool to derive this law from Ampere’s law and Gauss’s law for magnetostatics. In addition, we will study the boundary conditions at conductive media interfaces, and introduce the instantaneous Poynting’s theorem.
49 50 Electromagnetic Field Theory 5.1 Derivation of Biot-Savart Law
Figure 5.1: A current element used to illustrate the derivation of Biot-Savart law. The current element generates a magnetic field due to Ampere’s law in the static limit. This law was established experimentally, but here, we will derive this law based on our mathematical knowledge so far.
From Gauss’ law and Ampere’s law in the static limit, and using the definition of the Green’s function, we have derived that ¦ µ J(r0) A(r) = dV 0 (5.1.1) 4π V R where R = |r − r0|. When the current element is small, and is carried by a wire of cross sectional area ∆a as shown in Figure 5.1, we can approximate the integrand as
J(r0)dV 0 ≈ J(r0)∆V 0 = (∆a)∆l ˆlI/∆a (5.1.2) | {z } | {z } ∆V J(r0)
In the above, ∆V = (∆a)∆l and ˆlI/∆a = J(r0) since J has the unit of amperes/m2. Here, ˆl is a unit vector pointing in the direction of the current flow. Hence, we can let the current element be
J(r0)∆V 0 = I∆l0 (5.1.3) where the vector ∆l0 = ∆lˆl, and 0 indicates that it is located at r0. Therefore, the incremental vector potential due to an incremental current element J(r0)∆V 0 is
µ J(r0)∆V 0 µ I∆l0 ∆A(r) = = (5.1.4) 4π R 4π R Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem 51
Since B = ∇ × A, we derive that the incremental B flux, ∆B due to the incremental current I∆l0 is
µI ∆l0 −µI 1 ∆B = ∇ × ∆A(r) = ∇ × = ∆l0 × ∇ (5.1.5) 4π R 4π R where we have made use of the fact that ∇×af(r) = −a×∇f(r) when a is a constant vector. The above can be simplified further making use of the fact that1
1 1 ∇ = − Rˆ (5.1.6) R R2 where Rˆ is a unit vector pointing in the r − r0 direction. We have also made use of the fact that R = p(x − x0)2 + (y − y0)2 + (z − z0)2. Consequently, assuming that the incremental length becomes infinitesimally small, or ∆l → dl, we have, after using (5.1.6) in (5.1.5), that
µI 1 dB = dl0 × Rˆ 4π R2 µIdl0 × Rˆ = (5.1.7) 4πR2
Since B = µH, we have
Idl0 × Rˆ dH = (5.1.8) 4πR2 or
¢ I(r0)dl0 × Rˆ H(r) = (5.1.9) 4πR2 which is Biot-Savart law, first determined experimentally, now derived using electromagnetic field theory.
1This is best done by expressing the ∇ operator in spherical coordinates. 52 Electromagnetic Field Theory 5.2 Shielding by Conductive Media 5.2.1 Boundary Conditions–Conductive Media Case
Figure 5.2: The schematics for deriving the boundary condition for the current density J at the interface of two conductive media.
In a conductive medium, J = σE, which is just a statement of Ohm’s law. From the current continuity equation, which is derivable from Ampere’s law and Gauss’ law for electric flux, one gets ∂% ∇ · J = − (5.2.1) ∂t If the right-hand side is everywhere finite, it will not induce a jump discontinuity in the current. Moreover, it is zero for static limit. Hence, just like the Gauss’s law case, the above implies that the normal component of the current Jn is continuous, or that J1n = J2n in the static limit. In other words,
nˆ · (J2 − J1) = 0 (5.2.2) Hence, using J = σE, we have
σ2E2n − σ1E1n = 0 (5.2.3)
The above has to be always true in the static limit irrespective of the values of σ1 and σ2. But Gauss’s law implies the boundary condition that
ε2E2n − ε1E1n = %s (5.2.4)
The above equation is incompatible with (5.2.3) unless %s 6= 0. Hence, surface charge density or charge accumulation is necessary at the interface, unless σ2/σ1 = ε2/ε1. This is found in semiconductor materials which are both conductive and having a permitivity: interfacial charges appear at the interface of two semi-conductor materials. Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem 53
5.2.2 Electric Field Inside a Conductor The electric field inside a perfect electric conductor (PEC) has to be zero by the explanation as follows. If medium 1 is a perfect electric conductor, then σ → ∞ but J1 = σE1. An infinitesimal E1 will give rise to an infinite current J1. To avoid this ludicrous situation, E1 has to be 0. This implies that D1 = 0 as well.
Figure 5.3: The behavior of the electric field and electric flux at the interface of a perfect electric conductor and free space (or air).
Since tangential E is continuous, from Faraday’s law, it is still true that
E2t = E1t = 0 (5.2.5) orn ˆ × E = 0. But since
nˆ · (D2 − D1) = %s (5.2.6) and that D1 = 0, then
nˆ · D2 = %s (5.2.7)
So surface charge density has to be nonzero at a PEC/air interface. Moreover, normal D2 6= 0, tangential E2 = 0: Thus the E and D have to be normal to the PEC surface. The sketch of the electric field in the vicinity of a perfect electric conducting (PEC) surface is shown in Figure 5.3. The above argument for zero electric field inside a perfect conductor is true for electro- dynamic problems. However, one does not need the above argument regarding the shielding of the static electric field from a conducting region. In the situation of the two conducting 54 Electromagnetic Field Theory objects example below, as long as the electric fields are non-zero in the objects, currents will keep flowing. They will flow until the charges in the two objects orient themselves so that electric current cannot flow anymore. This happens when the charges produce internal fields that cancel each other giving rise to zero field inside the two objects. Faraday’s law still applies which means that tangental E field has to be continuous. Therefore, the boundary condition that the fields have to be normal to the conducting object surface is still true for elecrostatics. A sketch of the electric field between two conducting spheres is show in Figure 5.4.
Figure 5.4: The behavior of the electric field and flux outside two conductors in the static limit. The two conductors need not be PEC, and yet, the fields are normal to the interface.
5.2.3 Magnetic Field Inside a Conductor
We have seen that for a finite conductor, as long as σ 6= 0, the charges will re-orient themselves until the electric field is expelled from the conductor; otherwise, the current will keep flowing until E = 0 or ∂tE = 0. In a word, static E is zero inside a conductor. But there are no magnetic charges nor magnetic conductors in this world. Thus this physical phenomenon does not happen for magnetic field: In other words, static magnetic field cannot be expelled from an electric conductor. However, a magnetic field can be expelled from a perfect conductor or a superconductor. You can only fully understand this physical phenomenon if we study the time-varying form of Maxwell’s equations. In a perfect conductor where σ → ∞, it is unstable for the magnetic field B to be nonzero. As time varying magnetic field gives rise to an electric field by the time-varying form of Faraday’s law, a small time variation of the B field will give rise to infinite current flow in a perfect conductor. Therefore to avoid this ludicrous situation, and to be stable, B = 0 in a perfect conductor or a superconductor. Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem 55
So if medium 1 is a perfect electric conductor (PEC), then B1 = H1 = 0. The boundary condition for magnetic field from Ampere’s law
nˆ × (H2 − H1) =n ˆ × H2 = Js (5.2.8)
which is the jump condition for the magnetic field. In other words, a surface current Js has to flow at the surface of a PEC in order to support the jump discontinuity in the tangential component of the magnetic field. From Gauss’s law,n ˆ · B is always continuous at an interface because of the absence of magnetic charges. The magnetic flux B1 is expelled from the perfect conductor makingn ˆ · B1 = 0 zero both inside and outside the PEC (due to the continuity of normal B at an interface). And hence, there is no normal component of the B1 field at the interface. Therefore, the boundary condition for B2 becomes, for a PEC,
nˆ · B2 = 0 (5.2.9)
The B field in the vicinity of a perfect conductor surface is as shown in Figure 5.5.
Figure 5.5: Sketch of the magnetic flux B around a perfect electric conductor. It is seen that nˆ · B = 0 at the surface of the perfect conductor.
When a superconductor cube is placed next to a static magnetic field near a permanent magnet, eddy current will be induced on the superconductor. The eddy current will expel the static magnetic field from the permanent magnet, or in a word, it will produce a magnetic dipole on the superconducting cube that repels the static magnetic field. Since these two magnetic dipoles are of opposite polarity, they repel each other, and cause the superconducting cube to levitate on the static magnetic field as shown in Figure 5.6. 56 Electromagnetic Field Theory
Figure 5.6: Levitation of a superconducting disk on top of a static magnetic field due to expulsion of the magnetic field from the superconductor.. This is also known as the Meissner effect (figure courtesy of Wikimedia).
5.3 Instantaneous Poynting’s Theorem
Before we proceed further with studying energy and power, it is habitual to add fictitious magnetic current M and fictitious magnetic charge %m to Maxwell’s equations to make them symmetric mathematically.2 To this end, we have
∂B ∇ × E = − − M (5.3.1) ∂t ∂D ∇ × H = + J (5.3.2) ∂t ∇ · D = % (5.3.3)
∇ · B = %m (5.3.4)
Consider the first two of Maxwell’s equations where fictitious magnetic current is included and that the medium is isotropic such that B = µH and D = εE. Next, we need to consider only the first two equations (since in electrodynamics, by invoking charge conservation, the third and the fourth equations are derivable from the first two). They are
∂B ∂H ∇ × E = − − M = −µ − M (5.3.5) ∂t i ∂t i ∂D ∂E ∇ × H = + J = ε + J + σE (5.3.6) ∂t ∂t i where Mi and Ji are impressed current sources. They are sources that are impressed into the system, and they cannot be changed by their interaction with the environment [50]. Also, for a conductive medium, a conduction current or induced current flows in addition to impressed current. Here, J = σE is the induced current source in the conductor. Moreover, J = σE is similar to ohm’s law. We can show from By dot multiplying (5.3.5) with H, and
2Even though magnetic current does not exist, electric current can be engineered to look like magnetic current as shall be learnt. Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem 57 dot multiplying (5.3.6) with E, we can show that ∂H H · ∇ × E = −µH · − H · M (5.3.7) ∂t i ∂E E · ∇ × H = εE · + E · J + σE · E (5.3.8) ∂t i Using the identity, which is the same as the product rule for derivatives, we have3
∇ · (E × H) = H · (∇ × E) − E · (∇ × H) (5.3.9)
Therefore, from (5.3.7), (5.3.8), and (5.3.9) we have
∂H ∂E ∇ · (E × H) = − µH · + εE · + σE · E + H · M + E · J (5.3.10) ∂t ∂t i i
To elucidate the physical meaning of the above, we first consider σ = 0, and Mi = Ji = 0, or the absence of conductive loss and the impressed current sources. Then the above becomes ∂H ∂E ∇ · (E × H) = − µH · + εE · (5.3.11) ∂t ∂t Rewriting each term on the right-hand side of the above, we have ∂H 1 ∂ ∂ 1 ∂ µH · = µ H · H = µ|H|2 = W (5.3.12) ∂t 2 ∂t ∂t 2 ∂t m ∂E 1 ∂ ∂ 1 ∂ εE · = ε E · E = ε|E|2 = W (5.3.13) ∂t 2 ∂t ∂t 2 ∂t e where |H(r, t)|2 = H(r, t) · H(r, t), and |E(r, t)|2 = E(r, t) · E(r, t). Then (5.3.11) becomes ∂ ∇ · (E × H) = − (W + W ) (5.3.14) ∂t m e where 1 1 W = µ|H|2,W = ε|E|2 (5.3.15) m 2 e 2 Equation (5.3.14) is reminiscent of the current continuity equation, namely that, ∂% ∇ · J = − (5.3.16) ∂t which is a statement of charge conservation. In other words, time variation of current density at a point is due to charge density flow into or out of the point. The vector quantity
Sp = E × H (5.3.17)
3The cyclical identity that a · (b × c) = c · (a × b) = b · (c × a) is useful for the derivation. 58 Electromagnetic Field Theory is called the Poynting’s vector , and (5.3.14) becomes
∂ ∇ · S = − W (5.3.18) p ∂t t where Wt = We + Wm is the total energy density stored in the electric and magnetic fields. The above is similar to the current continuity equation in physical interpretation mentioned above. Analogous to that current density is charge density flow, power density is energy density flow. It is easy to show that Sp has a dimension of watts per meter square, or power density, and that Wt has a dimension of joules per meter cube, or energy density. Now, if we let σ 6= 0, then the term to be included is then σE · E = σ|E|2 which has the unit of S m−1 times V2 m−2, or W m−3 where S is siemens. We arrive at this unit by 1 V 2 noticing that 2 R is the power dissipated in a resistor of R ohms with a unit of watts. The reciprocal unit of ohms, which used to be called mhos is now called siemens. With σ 6= 0, (5.3.18) becomes
∂ ∂ ∇ · S = − W − σ|E|2 = − W − P (5.3.19) p ∂t t ∂t t d
2 Here, ∇·Sp has physical meaning of power density oozing out from a point, and −Pd = −σ|E| has the physical meaning of power density dissipated (siphoned) at a point by the conductive loss in the medium which is proportional to −σ|E|2.
Now if we set Ji and Mi to be nonzero, (5.3.19) is augmented by the last two terms in (5.3.10), or
∂ ∇ · S = − W − P − H · M − E · J (5.3.20) p ∂t t d i i
The last two terms can be interpreted as the power density supplied by the impressed currents Mi and Ji. Therefore, (5.3.20) becomes
∂ ∇ · S = − W − P + P (5.3.21) p ∂t t d s where
Ps = −H · Mi − E · Ji (5.3.22)
Here, Ps is the power supplied by the impressed current sources. These terms are positive if H and Mi have opposite signs, or if E and Ji have opposite signs. The last terms reminds us of what happens in a negative resistance device or a battery.4 In a battery, positive charges move from a region of lower potential to a region of higher potential (see Figure 5.7). The positive charges move from one end of a battery to the other end of the battery. Hence, they are doing an “uphill climb” driven by chemical processes within the battery.
4A negative resistance has been made by Leo Esaki [51], winning him a share in the Nobel prize. Biot-Savart law, Conductive Media Interface, Instantaneous Poynting’s Theorem 59
Figure 5.7: Figure showing the dissipation of energy as the current flows around a loop. A battery can be viewed as having a negative resistance.
−3 In the above, one can easily work out that Ps has the unit of W m which is power supplied density. One can also choose to rewrite (5.3.21) in integral form by integrating it over a volume V and invoking the divergence theorem yielding ¢ ¢ ¢ ¢ d dS · Sp = − WtdV − PddV + PsdV (5.3.23) S dt V V V The left-hand side is ¢ ¢
dS · Sp = dS · (E × H) (5.3.24) S S which represents the power flowing out of the surface S. 60 Electromagnetic Field Theory Lecture 6
Time-Harmonic Fields, Complex Power
The analysis of Maxwell’s equations can be greatly simplified by assuming the fields to be time harmonic, or sinusoidal (cosinusoidal). Electrical engineers use a method called phasor technique [32,52], to simplify equations involving time-harmonic signals. This is also a poor- man’s Fourier transform [53]. That is one begets the benefits of Fourier transform technique without knowledge of Fourier transform. Since only one time-harmonic frequency is involved, this is also called frequency domain analysis.1 Phasors are represented in complex numbers. From this we will also discuss the concept of complex power.
1It is simple only for linear systems: for nonlinear systems, such analysis can be quite unwieldy. But rest assured, as we will not discuss nonlinear systems in this course.
61 62 Electromagnetic Field Theory 6.1 Time-Harmonic Fields—Linear Systems
Figure 6.1: A commemorative stamp showing the contribution of Euler (courtesy of Wikipedia and Pinterest).
To learn phasor technique, one makes use the formula due to Euler (1707–1783) (Wikipedia) ejα = cos α + j sin α (6.1.1) √ where j = −1 is an imaginary number.2 From Euler’s formula one gets cos α = h jωti Ex(x, y, z, t) = Ex(r, t) = The above can be repeated for y and z components. Compactly, one can write h i E(r, t) = ∂B ∇ × E = − − M (6.1.12) ∂t assuming that all quantities are time harmonic, then h i E(r, t) = In the phasor representation, Maxwell’s equations has no time derivatives; hence the equations are simplified. We can also arrive at the above simplified equations using Fourier transform technique. To this end, we use Faraday’s law as an example. By letting ¢ ∞ 1 E(r, t) = E(r, ω)ejωtdω (6.2.1) 2π −∞ ¢ ∞ 1 B(r, t) = B(r, ω)ejωtdω (6.2.2) 2π −∞ ¢ ∞ 1 M(r, t) = M(r, ω)ejωtdω (6.2.3) 2π −∞ Substituting the above into Faraday’s law given by (6.1.12), we get ¢ ∞ ¢ ∞ ¢ ∞ ∂ ∇ × dωejωtE(r, ω) = − dωejωtB(r, ω) − dωejωtM(r, ω) (6.2.4) ∂t −∞ −∞ −∞ Using the fact that ¢ ∞ ¢ ∞ ¢ ∞ ∂ ∂ dωejωtB(r, ω) = dω ejωtB(r, ω) = dωejωtjωB(r, ω) (6.2.5) ∂t ∂t −∞ −∞ −∞ and that ¢ ∞ ¢ ∞ ∇ × dωejωtE(r, ω) = dωejωt∇ × E(r, ω) (6.2.6) −∞ −∞ Time-Harmonic Fields, Complex Power 65 Furthermore, using the fact that ¢ ∞ ¢ ∞ dωejωtA(ω) = dωejωtB(ω), ∀t (6.2.7) −∞ −∞ implies that A(ω) = B(ω), and using (6.2.5) and (6.2.6) in (6.2.4), and the property (6.2.7), one gets ∇ × E(r, ω) = −jωB(r, ω) − M(r, ω) (6.2.8) These equations look exactly like the phasor equations we have derived previously, save that the field E(r, ω), B(r, ω), and M(r, ω) are now the Fourier transforms of the field E(r, t), B(r, t), and M(r, t). Moreover, the Fourier transform variables can be complex just like phasors. Repeating the exercise above for the other Maxwell’s equations, we obtain equations that look similar to those for their phasor representations. Hence, Maxwell’s equations can be simplified either by using phasor technique or Fourier transform technique. However, the dimensions of the phasors are different from the dimensions of the Fourier-transformed fields: E(r) and E(r, ω) do not have the same dimension on closer examination. e 6.3 Complex Power Consider now that in the phasor representations, E(r) and H(r) are complex vectors, and their cross product, E(r) × H∗(r), which still has thee unit ofe power density, has a different physical meaning. First,e considere the instantaneous Poynting’s vector S(r, t) = E(r, t) × H(r, t) (6.3.1) where all the quantities are real valued. Now, we can use phasor technique to analyze the above. Assuming time-harmonic fields, the above can be rewritten as h i h i S(r, t) = 1 h i S = hS(r, t)i = S = E × H∗ (6.3.8) e e e is termed the complex Poynting’s vector . The power flow associated with it is termed complex power. Figure 6.2: A simple circuit example to illustrate the concept of complex power in circuit theory. To understand what complex power is , it is fruitful if we revisit complex power [50, 54] in our circuit theory course. The circuit in Figure 6.2 can be easily solved by using phasor technique. The impedance of the circuit is Z = R + jωL. Hence, V = (R + jωL)I (6.3.9) e e where V and I are the phasors of the voltage and current for time-harmonic signals. Just as in the electromagnetice e case, the complex power is taken to be P = V I∗ (6.3.10) e ee But the instantaneous power is given by Pinst(t) = V (t)I(t) (6.3.11) Time-Harmonic Fields, Complex Power 67 where V (t) = h jωti V (t) = V0 cos(ωt) = h jωti I(t) = I0 cos(ωt + α) = = V0I0 cos(ωt) [cos(ωt) cos(α) − sin(ωt) sin α] 2 = V0I0 cos (ωt) cos α − V0I0 cos(ωt) sin(ωt) sin α (6.3.15) It can be seen that the first term does not time-average to zero, but the second term, by letting cos(ωt) sin(ωt) = 0.5 sin(2ωt), does time-average to zero. Now taking the time average of (6.3.15), we get 1 1 h i P = hP i = V I cos α = 4Because that complex power is proportional to V I∗, it is the relative phase between V and I that matters. Therefore, α above is the relative phase between theee phasor current and phasor voltage.e e 68 Electromagnetic Field Theory to our home, the power is complex because the current and voltage are not in phase. Even though the reactive power time-averages to zero, the power company still needs to deliver it to and from our home to run our washing machine, dish washer, fans, and air conditioner etc, and hence, charges us for it. Part of this power will be dissipated in the transmission lines that deliver power to our home. In other words, we have to pay to the use of imaginary power! Figure 6.3: Plots of instantaneous power for when the voltage and the current is in phase (α = 0), and when they are out of phase (α 6= 0). In the out-of-phase case, there is an additional time-varying term that does not contribute to time-average power as shown in (6.3.15). Lecture 7 More on Constitute Relations, Uniform Plane Wave As mentioned before, constitutive relations are important for us to solve only the first two of four Maxwell’s equations. Assuming that J is known or zero, then the first two vector equations have four vector unknowns: E, H, D, and B, which cannot be determined by solving only two equations. The addition two equations come from te constitutive relations. Constitutive relations are useful because they allow us to incorporate material properties into the solutions of Maxwell’s equations. The material properties can be frequency dispersive, anisotropic, bi-anisotropic, inhomogeneous, lossy, conductive, nonlinear as well as spatially dispersive. The use of phasors or frequency domain method will further simplify the charac- terization of different media., The use of phasors allows us to study wave phenomena in these media quite easily. Hence, we will also study uniform plane wave in such media, including a lossy conductive media. 7.1 More on Constitutive Relations As have been said, Maxwell’s equations are not solvable until the constitutive relations are included. Here, we will look into depth more into various kinds of constitutive relations. Now that we have learned phasor technique, we can study a more general constitutive relationship compared to what we have seen earlier. 7.1.1 Isotropic Frequency Dispersive Media First let us look at the simple linear constitutive relation previously discussed for dielectric media where [31], [32][p. 82], [48] D = ε0E + P (7.1.1) 69 70 Electromagnetic Field Theory We have a simple model where P = ε0χ0E (7.1.2) where χ0 is the electric susceptibility. When used in the generalized Ampere’s law, P, the polarization density, plays an important role for the flow of the displacement current through space. The polarization density is due to the presence of polar atoms or molecules that become little dipoles in the presence of an electric field. For instance, water, which is H2O, is a polar molecule that becomes a small dipole when an electric field is applied. We can think of displacement current flow as capacitive coupling between the dipoles yielding polarization current flow through space. Namely, for a source-free medium, ∂D ∂E ∂P ∇ × H = = ε + (7.1.3) ∂t 0 ∂t ∂t Figure 7.1: As a series of dipoles line up end to end, one can see a current flowing through the line of dipoles as they oscillate back and forth in their polarity. This is similar to how displacement current flows through a capacitor. For example, for a sinusoidal oscillating field, the dipoles will flip back and forth giving rise to flow of displacement current just as how time-harmonic electric current can flow through a capacitor as shown in Figure 7.1. The linear relationship above can be generalized to that of a linear time-invariant system [52], or that at any given r [35][p. 212], [48][p. 330]. P(r, t) = ε0χe(r, t) ~ E(r, t) (7.1.4) where ~ here implies a convolution. In the frequency domain or the Fourier space, the above linear relationship becomes P(r, ω) = ε0χ0(r, ω)E(r, ω), (7.1.5) or D(r, ω) = ε0 [1 + χ0(r, ω)] E(r, ω) = ε(r, ω)E(r, ω) (7.1.6) where ε(r, ω) = ε0 [1 + χ0(r, ω)] at any point r in space. There is a rich variety of ways at which χ0(ω) can manifest itself. Such a permittivity ε(r, ω) is often called the effective permittivity. Such media where the effective permittivity is a function of frequency is termed dispersive media, or frequency dispersive media. More on Constitute Relations, Uniform Plane Wave 71 7.1.2 Anisotropic Media For anisotropic media [32][p. 83] D = ε0E + ε0χ0(ω) · E = ε0 I + χ0(ω) · E = ε(ω) · E (7.1.7) In the above, ε is a 3×3 matrix also known as a tensor in electromagnetics. The above implies that D and E do not necessary point in the same direction: the meaning of anisotropy. (A tensor is often associated with a physical notion like the relation between two physical fields, whereas a matrix is not.) Previously, we have assume that χ0 to be frequency independent. This is not usually the case as all materials have χ0’s that are frequency dependent. (This will become clear later.) Also, since ε(ω) is frequency dependent, we should view it as a transfer function where the input is E, and the output D. This implies that in the time-domain, the above relation becomes a time-convolution relation as in (7.1.4). Similarly for conductive media, J = σE, (7.1.8) This can be used in Maxwell’s equations in the frequency domain to yield the definition of complex permittivity. Using the above in Ampere’s law in the frequency domain, we have ∇ × H(r) = jωεE(r) + σE(r) = jωε(ω)E(r) (7.1.9) e where the complex permittivity ε(ω) = ε − jσ/ω. Notice that Ampere’s law in the frequency domain with complex permittivitye in (7.1.9) is no more complicated than Ampere’s law for nonconductive media. The algebra for complex numbers is no more difficult than the algebra for real numbers. This is one of the strengths of phasor technique. For anisotropic conductive media, one has J = σ(ω) · E, (7.1.10) Here, again, due to the tensorial nature of the conductivity σ, the electric current J and electric field E do not necessary point in the same direction. The above assumes a local or point-wise relationship between the input and the output of a linear system. This need not be so. In fact, the most general linear relationship between P(r, t) and E(r, t) is ¢ ¦ ∞ ∞ P(r, t) = χ(r − r0, t − t0) · E(r0, t0)dr0dt0 (7.1.11) −∞ −∞ The above is a general convolutional relationship in both space and time. In the Fourier transform space, by taking Fourier transform in both space and time, the above becomes P(k, ω) = χ(k, ω) · E(k, ω) (7.1.12) 72 Electromagnetic Field Theory where ¢ ∞ χ(k, ω) = χ(r, t) exp(jk · r − jωt)drdt (7.1.13) −∞ (The dr integral above is actually a three-fold integral with dr = dxdydz.) Such a medium is termed spatially dispersive as well as frequency dispersive [35][p. 6], [55]. In general1 ε(k, ω) = 1 + χ(k, ω) (7.1.14) where D(k, ω) = ε(k, ω) · E(k, ω) (7.1.15) The above can be extended to magnetic field and magnetic flux yielding B(k, ω) = µ(k, ω) · H(k, ω) (7.1.16) for a general spatial and frequency dispersive magnetic material. In optics, most materials are non-magnetic, and hence, µ = µ0, whereas it is quite easy to make anisotropic magnetic materials in radio and microwave frequencies, such as ferrites. 7.1.3 Bi-anisotropic Media In the previous section, the electric flux D depends on the electric field E and the magnetic flux B, on the magnetic field H. The concept of constitutive relation can be extended to where D and B depend on both E and H. In general, one can write D = ε(ω) · E + ξ(ω) · H (7.1.17) B = ζ(ω) · E + µ(ω) · H (7.1.18) A medium where the electric flux or the magnetic flux is dependent on both E and H is known as a bi-anisotropic medium [32][p. 81]. 7.1.4 Inhomogeneous Media If any of the ε, ξ, ζ, or µ is a function of position r, the medium is known as an inhomogeneous medium or a heterogeneous medium. There are usually no simple solutions to problems associated with such media [35]. 7.1.5 Uniaxial and Biaxial Media Anisotropic optical materials are often encountered in optics. Examples of them are the biaxial and uniaxial media, and discussions of them are often found in optics books [56–58]. They are optical materials where the permittivity tensor can be written as ε1 0 0 ε = 0 ε2 0 (7.1.19) 0 0 ε3 1In the following, one should replace the 1 with an identity operator, but it is generally implied. More on Constitute Relations, Uniform Plane Wave 73 When ε1 6= ε2 6= ε3, the medium is known as a biaxial medium. But when ε1 = ε2 6= ε3, then the medium is a uniaxial medium. In the biaxial medium, all three components of the electric field feel different permittivity constants. But in the uniaxial medium, the electric field in the xy plane feels the same permittivity constant, but the electric field in the z direction feels a different permittivity constant. As shall be shown later, different light polarization will propagate with different behavior through such a medium. 7.1.6 Nonlinear Media In the previous cases, we have assumed that χ0 is independent of the field E. The relationships between P and E can be written more generally as P = ε0χ0(E) · E (7.1.20) where the relationship can appear in many different forms. For nonlinear media, the rela- tionship can be nonlinear as indicated in the above. It can be easily shown that the principle of linear superposition does not hold for the above equation, a root test of linearity. Non- linear permittivity effect is important in optics. Here, the wavelength is short, and a small change in the permittivity or refractive index can give rise to cumulative phase delay as the wave has to propagate many wavelengths through a nonlinear optical medium [59–61]. Kerr optical nonlinearity, discovered in 1875, was one of the earliest nonlinear phenomena observed [32, 56,59]. For magnetic materials, nonlinearity can occur in the effective permeability of the medium. In other words, B = µ(H) · H (7.1.21) This nonlinearity is important even at low frequencies, and in electric machinery designs [62, 63], and magnetic resonance imaging systems [64]. The large permeability in magnetic materials is usually due to the formation of magnetic domains which can only happen at low frequencies. 7.2 Wave Phenomenon in the Frequency Domain We have seen the emergence of wave phenomenon in the time domain. Given the simplicity of the frequency domain method, it will be interesting to ask how this phenomenon presents itself for time-harmonic field or in the frequency domain. In the frequency domain, the source-free Maxwell’s equations are [32][p. 429], [65][p. 107] ∇ × E(r) = −jωµ0H(r) (7.2.1) ∇ × H(r) = jωε0E(r) (7.2.2) Taking the curl of (7.2.1) and then substituting (7.2.2) into its right-hand side, one obtains 2 ∇ × ∇ × E(r) = −jωµ0∇ × H(r) = ω µ0ε0E(r) (7.2.3) 74 Electromagnetic Field Theory Again, using the identity that ∇ × ∇ × E = ∇(∇ · E) − ∇ · ∇E = ∇(∇ · E) − ∇2E (7.2.4) and that ∇ · E = 0 in a source-free medium, (7.2.3) becomes 2 2 (∇ + ω µ0ε0)E(r) = 0 (7.2.5) This is known as the Helmholtz wave equation or just the Helmholtz equation.2 For simplicity of seeing the wave phenomenon, we let E =xE ˆ x(z), a field pointing in the x direction, but varying only in the z direction. Evidently, ∇ · E(r) = ∂Ex(z)/∂x = 0. Then with ∂/∂x = 0 and ∂/∂y = 0, (7.2.5) simplifies to d2 + k2 E (z) = 0 (7.2.6) dz2 0 x 2 2 2 2 √ where k0 = ω µ0ε0 = ω /c0 where c0 = 1/ µ0ε0 is the velocity of light. The general solution to (7.2.6) is of the form −jk0z jk0z Ex(z) = E0+e + E0−e (7.2.7) One can convert the above back to the time domain using phasor technique, or by using that jωt Ex(z, t) = Ex(z, t) = |E0+| cos(ωt − k0z + α+) + |E0−| cos(ωt + k0z + α−) (7.2.8) where we have assumed that jα± E0± = |E0±|e (7.2.9) The physical picture of the above expressions can be appreciated by rewriting ω cos(ωt ∓ k0z + α±) = cos (c0t ∓ z) + α± (7.2.10) c0 ω where we have used the fact that k0 = . One can see that the first term on the right-hand c0 side of (7.2.8) is a sinusoidal plane wave traveling to the right, while the second term is a sinusoidal plane wave traveling to the left, both with velocity c0. The above plane wave is uniform and a constant in the xy plane and propagating in the z direction. Hence, it is also called a uniform plane wave in 1D. Moreover, for a fixed t or at t = 0, the sinusoidal functions are proportional to cos(∓k0z + α±). This is a periodic function in z with period 2π/k0 which is the wavelength λ0, or that 2π ω 2πf k0 = = = (7.2.11) λ0 c0 c0 One can see that because c0 is a humongous number in electromagnetics, λ0 can be very large. You can plug in the frequency of your local AM 920 station, operating at 920 KHz, to see how big λ0 is. The above analysis still holds true if and µ are replaced by and µ but are real 0 0 √ numbers. In this case, the velocity c of the wave is now given by c = 1/ µ. This velocity is the velocity of the phase of a time-harmonic signal and hence, is known as phase velocity. 2For a comprehensive review of this topic, one may read the lecture notes [44]. More on Constitute Relations, Uniform Plane Wave 75 7.3 Uniform Plane Waves in 3D By repeating the previous derivation for a homogeneous lossless medium, the vector Helmholtz equation for a source-free medium is given by [44] ∇ × ∇ × E − ω2µεE = 0 (7.3.1) By the same derivation as before for the free-space case, since ∇ · E = 0 due to source-free medium, one has ∇2E + ω2µεE = 0 (7.3.2) The general solution to (7.3.2) is hence −jkxx−jky y−jkz z −jk·r E = E0e = E0e (7.3.3) where k =xk ˆ x +yk ˆ y +zk ˆ z, r =xx ˆ +yy ˆ +zz ˆ and E0 is a constant vector. And upon substituting (7.3.3) into (7.3.2), it is seen that 2 2 2 2 kx + ky + kz = ω µε (7.3.4) This is called the dispersion relation for a plane wave. In general, kx, ky, and kz can be arbitrary and even complex numbers as long as this relation is satisfied. To simplify the discussion, we will focus on the case where kx, ky, and kz are all real numbers. When this is the case, the vector function in (7.3.3) represents a uniform plane wave propagating in the k direction. As can be seen, when k · r = constant, it is represented by all points of r that represents a flat plane (see Figure 7.2). This flat plane represents the constant phase wave front. By increasing the constant, we obtain different planes for progressively changing phase fronts.3 Figure 7.2: A figure showing the geometrical meaning of k · r equal to a constant. It is a flat plane that defines the wavefront of a plane wave. 3In the exp(jωt) time convention, this phase front is decreasing, whereas in the exp(−iωt) time convention, this phase front is increasing. 76 Electromagnetic Field Theory Further, since ∇ · E = 0, we have −jkxx−jky y−jkz z −jk·r ∇ · E = ∇ · E0e = ∇ · E0e −jk·r = (−xjkˆ x − yjkˆ y − zjkˆ z) · E0e = −j(ˆxkx +yk ˆ y +zk ˆ z) · E = 0 (7.3.5) or that k · E0 = k · E = 0 (7.3.6) Thus, E is orthogonal to k for a uniform plane wave. The above exercise shows that whenever E is a plane wave, and when the ∇ operator operates on such a vector function, one can do the substitution that ∇ → −jk. Hence, in a source-free homogenous medium, ∇ × E = −jωµH (7.3.7) the above equation simplifies to −jk × E = −jωµH (7.3.8) or that k × E H = (7.3.9) ωµ Similar to (7.3.3), we can define −jkxx−jky y−jkz z −jk·r H = H0e = H0e (7.3.10) Then using (7.3.3) in (7.3.9), it is clear that k × E H = 0 (7.3.11) 0 ωµ We can assume that E0 and H0 are real vectors. Then E0, H0 and k form a right-handed system, or that E0 ×H0 point in the direction of k. (This also implies that E, H and k form a right-handed system.) Such a wave, where the electric field and magnetic field are transverse to the direction of propagation, is called a transverse electromagnetic (TEM) wave. Figure 7.3: The E, H, and k together form a right-hand coordinate system, obeying the right-hand rule. Namely, E × H points in the direction of k. More on Constitute Relations, Uniform Plane Wave 77 Figure 7.3 shows that k · E = 0, and that k × E points in the direction of H as shown in (7.3.9). Figure 7.3 also shows, as k, E, and H are orthogonal to each other. Since in general, E0 and H0 can be complex vectors, because they are phasors, we need to show the more general case. From (7.3.9), one can show, using the “back-of-the-cab” formula, that k k E × H∗ = E · E∗ = |E|2 (7.3.12) ωµ ωµ But E × H∗ is the direction of power flow, and it is in fact in the k direction assuming that k is a real vector. This is also required by the Poynting’s theorem. Furthermore, we can show in general that |k||E| r ε 1 |H| = = |E| = |E| (7.3.13) ωµ µ η where the quantity rµ η = (7.3.14) ε is call the intrinsic impedance. For vacuum or free-space, it is about 377 Ω ≈ 120π Ω. Notice that the above analysis holds true as long as ε is real, and it can be frequency dispersive, since we are considering a mono-chromatic or time-harmonic field. Besides, for a mono-chromatic signal, the analysis in Section 7.2 still applies except that the velocity of √ light is now given by c = 1/ µ0ε. As we shall see, this velocity is the phase velocity of the mono-chromatic wave. In the above, when kx, ky, and kz are not all real, the wave is known as an inhomogeneous wave.4 4The term inhomogeneous plane wave is used sometimes, but it is a misnomer since there is no more a planar wave front in this case. 78 Electromagnetic Field Theory Lecture 8 Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model In the previous lecture, we see the power of phasor technique or the frequency domain analysis. The analysis of a frequency dispersive medium where ε is frequency dependent, is similar to that of free space or vacuum. The two problems are mathematically homomorphic to each other. In this lecture, we will generalize to the case where ε becomes a complex number, called the complex permittivity. Using phasor technique, this way of solving Maxwell’s equations is still homomorphic to that of solving Maxwell’s equations in free space. The analysis is greatly simplified as a result! 8.1 Plane Waves in Lossy Conductive Media Previously, we have derived the plane wave solution for a lossless homogeneous medium. Since the algebra of complex numbers is similar to that of real numbers,the derivation can be generalized to a conductive medium by invoking mathematical homomorphism. In other words, in a conductive, one only needs to replace the permittivity with a complex permittivity, as repeated here. When conductive loss is present, σ 6= 0, and J = σE. Then generalized Ampere’s law becomes σ ∇ × H = jωεE + σE = jω ε + E (8.1.1) jω σ A complex permittivity can be defined as ε = ε − j ω . Eq. (8.1.1) can be rewritten as e ∇ × H = jωεE (8.1.2) e 79 80 Electromagnetic Field Theory This equation is of the same form as source-free Ampere’s law in the frequency domain for a lossless medium where ε is completely real. Using the same method as before, a wave solution −jk·r E = E0e (8.1.3) will have the dispersion relation which is now given by 2 2 2 2 kx + ky + kz = ω µε (8.1.4) e Since ε is complex now, kx, ky, and kz cannot be all real. Equation (8.1.4) has been derived previouslye by assuming that k is a real vector. When k = k0 − jk00 is a complex vector, some of the previous derivations for real k vector may not be correct here for complex k vector. It is also difficult to visualize a complex k vector that is suppose to indicate the direciton with which the wave is propagating. Here, the wave can decay and oscillate in different directions. So again, for physical insight, we look at the simplified case where E =xE ˆ x(z) (8.1.5) √ so that ∇ · E = ∂xEx(z) = 0, and let k =zk ˆ =zω ˆ µε. This wave is constant in the xy plane, and hence, is a plane wave. Furthermore, in thise manner, we are requiring that the wave decays and propagates (or oscillates) only in the z direction. For such a simple plane wave, −jkz E =x ˆEx(z) =xE ˆ 0e (8.1.6) √ where k = ω µε, since k · k = k2 = ω2µε is still true. Faraday’s lawe gives rise to e s k × E kE (z) ε H = =y ˆ x =y ˆ E (z) (8.1.7) ωµ ωµ µe x √ where k vector is defined shortly after (8.1.5) above, and k = ω µε, a complex number. It is seen that H =yH ˆ y, and that e sµ E /H = (8.1.8) x y ε e 8.1.1 Highly Conductive Case σ σ When the medium is highly conductive, σ → ∞, and ε = −j ω ≈ −j ω . In other words, when σ | ω | ε, the conduction current dominates over displacemente current. Thus, the following approximation can be made, namely, r jσ k = ωqµε ' ω −µ = p−jωµσ (8.1.9) ω e Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 81 √ Taking −j = √1 (1 − j), we have for a highly conductive medium that 2 rωµσ k ' (1 − j) = k0 − jk00 (8.1.10) 2 For a plane wave, e−jkz, it then becomes 0 00 e−jkz = e−jk z−k z (8.1.11) By converting the above phasor back to the time domain, this plane wave decays exponentially as well as oscillates in the z direction. The reason being that a conductive medium is lossy, and it absorbs energy from the plane wave. This is similar to resistive loss we see in the resistive circuit. The penetration depth of this wave is then 1 r 2 δ = = (8.1.12) k00 ωµσ This distance δ, the penetration depth, is called the skin depth of a plane wave propagating in a highly lossy conductive medium where conduction current dominates over displacement current, or that σ ωε. This happens for radio wave propagating in the saline solution of the ocean, the Earth, or wave propagating in highly conductive metal, like your induction cooker. 8.1.2 Lowly Conductive Case When the conductivity is low, namely, when the displacement current is larger than the σ conduction current, then ωε 1, we have s r σ jσ k = ω µ ε − j = ω µε 1 − ω ωε √ 1 σ ≈ ω µε 1 − j = k0 − jk00 (8.1.13) 2 ωε The above is the approximation to k = k0 − jk00 for a low conductivity medium where 1 σ conduction current is much smaller than displacement current. The term ωε is called the loss tangent of a lossy medium. It is the ratio of the conduction current to the displacement current in a lossy conductive medium. In general, in a lossy medium ε = ε0 − jε00, and ε00/ε0 is called the loss tangent of the medium. It is to be noted that in the optics and physics community, e−iωt time convention is preferred. In that case, we need to do the switch j → −i, and a loss medium is denoted by ε = ε0 + iε00. 1We have made use of the approximation that (1 + x)n ≈ 1 + nx when x is small, which can be justified by Taylor series expansion. 82 Electromagnetic Field Theory 8.2 Lorentz Force Law The Lorentz force law is the generalization of the Coulomb’s law for forces between two charges. Lorentz force law includes the presence of a magnetic field. It is given by F = qE + qv × B (8.2.1) The first term on the right-hand side is the electric force similar to the statement of Coulomb’s law, while the second term is the magnetic force called the v × B force. This law can be also written in terms of the force density f which is the force on the charge density, instead of force on a single charge. By so doing, we arrive at f = %E + %v × B = %E + J × B (8.2.2) where % is the charge density, and one can identified the current J = %v. Lorentz force law can also be derived from the integral form of Faraday’s law, if one assumes that the law is applied to a moving loop intercepting a magnetic flux [66]. In other words, Lorentz force law and Faraday’s law are commensurate with each other. 8.3 Drude-Lorentz-Sommerfeld Model In the previous lecture, we have seen how loss can be introduced by having a conduction current flowing in a medium. Now that we have learnt the versatility of the frequency domain method and phasor technique, other loss mechanism can be easily introduced. First, let us look at the simple constitutive relation where D = ε0E + P (8.3.1) We have a simple model where P = ε0χE (8.3.2) where χ is the electric susceptibility. To see how χ(ω) can be derived, we will study the Drude-Lorentz-Sommerfeld model. This is usually just known as the Drude model or the Lorentz model in many textbooks although Sommerfeld also contributed to it. These models, the Drude and Lorentz models, can be unified in one equation as shall be shown. Figure 8.1: Polarization of an atom in the presence of an electric field. Here, it is assumed that the electron is weakly bound or unbound to the nucleus of the atom. Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 83 8.3.1 Cold Collisionless Plasma Medium We can first start with a simple electron driven by an electric field E in the absence of a magnetic field B.2 If the electron is free to move, then the force acting on it, from the Lorentz force law, is just −eE where q = −e is the charge of the electron (see Figure 8.1). Then from Newton’s law, assuming a one dimensional case, it follows that d2x m = −eE (8.3.3) e dt2 where the left-hand side is due to the inertial force of the mass of the electron, and the right-hand side is the electric force acting on a charge of −e coulomb. Here, we assume that E points in the x-direction, and we neglect the vector nature of the electric field or that we assume that both x and E are in the same direction. Writing the above in the frequency domain for time-harmonic fields, and using phasor technique, one gets 2 −ω mex = −eE (8.3.4) 2 The above implies that the inertial force of the electron, given by −ω mex, is of the same polarity as the electric field force on the electron which is −eE. From this, we have e x = 2 E (8.3.5) ω me implying that the displacement x is linearly proportional to the electric field amplitude E, or they are in phase. This, for instance, can happen in a plasma medium where the atoms are ionized, and the electrons are free to roam [67]. Hence, we assume that the positive ions are more massive, sluggish, and move very little compared to the electrons when an electric field is applied. The dipole moment formed by the displaced electron away from the ion due to the electric field is then e2 p = −ex = − 2 E (8.3.6) ω me for one electron. When there are N electrons per unit volume, the dipole moment density is then given by Ne2 P = Np = − 2 E (8.3.7) ω me In general, P and E point in the opposite directions, and we can write 2 2 Ne ωp P = − 2 E = − 2 ε0E (8.3.8) ω me ω 2Even if B 6= 0, the v × B force is small if the velocity of the electron is much smaller than the speed of light. 84 Electromagnetic Field Theory 2 2 where we have defined ωp = Ne /(meε0). Then, ω 2 D = ε E + P = ε 1 − p E (8.3.9) 0 0 ω2 In this manner, we see that the effective permittivity is ω 2 ε(ω) = ε 1 − p (8.3.10) 0 ω2 What the above math is saying is that the electric field E induces a dipole moment density P that is negative to ε0E, or the vacuum part of the contribution to D. This negative dipole density cancels the contribution to the electric flux from the vacuum 0E. For low frequency, the effective permittivity is negative, disallowing the propagation of a wave as we shall see. Hence, ε < 0 if p ω < ωp = N/(meε0)e √ Here, ωp is the plasma frequency. Since k = ω µε, if ε is negative, k = −jα becomes pure imaginary, and a wave such as e−jkz decays exponentially as e−αz. This is also known as an evanescent wave. In other words, the wave cannot propagate through such a medium: Our ionosphere is such a medium. The plasma shields out electromagnetic waves that are below the plasma frequency ωp. Therefore, it was extremely fortuitous that Marconi, in 1901, was able to send a radio signal from Cornwall, England, to Newfoundland, Canada. Nay sayers thought his experiment would never succeed as the radio signal would propagate to outer space and never to return. Fortunately so, it is the presence of the ionosphere that bounces the radio wave back to Earth, making his experiment a resounding success and a very historic one! Serendipity occurs in science and technology development more than once: the experiment also heralds in the age of wireless communications. This experiment also stirred interests into research on the ionosphere. It was an area again where Oliver Heaviside made contributions; as a result, a layer of the ionosphere is named Heaviside layer or Kennelly-Heaviside layer [68]. If you listen carefully to the broadway musical “Cats” by Andrew Lloyd Weber, there is a mention about the Heaviside layer in one of the verses! Figure 8.2: The electron is bound to the ion by an attractive force. This can be approximately modeled by a spring providing a restoring force to the electron. Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 85 8.3.2 Bound Electron Case The above model can be generalized to the case where the electron is bound to the ion, but the ion now provides a restoring force similar to that of a spring (see Figure 8.2), namely, d2x m + κx = −eE (8.3.11) e dt2 We assume that the ion provides a restoring force just like Hooke’s law. Again, for a time- harmonic field, (8.3.11) can be solved easily in the frequency domain to yield e e x = 2 E = 2 2 E (8.3.12) (ω me − κ) (ω − ω0 )me 2 where we have defined ω0 me = κ. The above is the typical solution of a lossless harmonic oscillator (pendulum) driven by an external force, in this case the electric field. The dipole moment due to an electric field then is e2 p = −ex = − 2 2 E (8.3.13) (ω − ω0 )me Therefore, when the frequency is low or ω = 0, this dipole moment is of the same polarity as the applied electric field E, contributing to a positive dipole moment. It contributes positively to the displacement flux D via P. 8.3.3 Damping or Dissipation Case Equation (8.3.11) can be generalized to the case when frictional, damping, or dissipation forces are present, or that d2x dx m + m Γ + κx = −eE (8.3.14) e dt2 e dt The second term on the left-hand side is a force that is proportional to the velocity dx/dt of the electron. This is the hall-mark of frictional force. Frictional force is due to the collision of the electrons with the background ions or lattice. It is proportional to the destruction (or dx change) of momentum (me dt ) of an electron. In the average sense, the destruction of the momentum is given by the product of the collision frequency and the momentum. In the above, Γ has the unit of frequency, and for plasma, and conductor, it can be regarded as a collision frequency. Solving the above in the frequency domain, one gets e x = 2 2 E (8.3.15) (ω − jωΓ − ω0 )me Following the same procedure in arriving at (8.3.7), we get −Ne2 P = 2 2 E (8.3.16) (ω − jωΓ − ω0 )me 86 Electromagnetic Field Theory In this, one can identify that −Ne2 χ(ω) = 2 2 (ω − jωΓ − ω0 )meε0 2 ωp = − 2 2 (8.3.17) ω − jωΓ − ω0 where ωp is as defined before. A function with the above frequency dependence is also called a Lorentzian function. It is the hallmark of a damped harmonic oscillator. If Γ = 0, then when ω = ω0, one sees an infinite resonance peak exhibited by the DLS model. But in the real world, Γ 6= 0, and when Γ is small, but ω ≈ ω0, then the peak value of χ is ω 2 ω 2 χ ≈ + p = −j p (8.3.18) jωΓ ωΓ χ exhibits a large negative imaginary part, the hallmark of a dissipative medium, as in the conducting medium we have previously studied. 8.3.4 Broad Applicability of Drude-Lorentz-Sommerfeld Model The DLS model is a wonderful model because it can capture phenomenologically the essence of the physics of many electromagnetic media, even though it is a purely classical model.3 It captures the resonance behavior of an atom absorbing energy from light excitation. When the light wave comes in at the correct frequency, it will excite electronic transition within an atom which can be approximately modeled as a resonator with behavior similar to that of a pendulum oscillator. This electronic resonances will be radiationally damped [34],4 and the damped oscillation can be modeled by Γ 6= 0. By picking a mixture of multi-species DLS oscillators, almost any shape of absorption spectra can be curve-fitted [69] (see Figure 8.3). 3What we mean here is that only Newton’s law has been used, and no quantum theory as yet. 4The oscillator radiates as it oscillates, and hence, loses energy to its environment. This causes the decay of the oscillation, just as a damped LC tank circuit losing energy to the resistor. Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 87 Figure 8.3: A Lorentzian has almost a bell-shape curve. By assuming multi-species of DLS oscillators in a medium, one can fit absorption spectra of almost any shape (courtesy of Wikipedia [69]). Moreover, the above model can also be used to model molecular vibrations. In this case, the mass of the electron will be replaced by the mass of the atom involved. The damping of the molecular vibration is caused by the hindered vibration of the molecule due to interaction with other molecules [70]. The hindered rotation or vibration of water molecules when excited by microwave is the source of heat in microwave heating. In the case of plasma, Γ 6= 0 represents the collision frequency between the free electrons and the ions, giving rise to loss. In the case of a conductor, Γ represents the collision frequency between the conduction electrons in the conduction band with the lattice of the material.5 Also, if there is no restoring force, then ω0 = 0. This is true for sea of electron moving in the conduction band of a medium. Besides, for sufficiently low frequency, the inertial force can be ignored. Thus, from (8.3.17), again we have6 ω 2 χ ≈ −j p (8.3.19) ωΓ and ω 2 ε = ε (1 + χ) = ε 1 − j p (8.3.20) 0 0 ωΓ We recall that for a conductive medium, we define a complex permittivity to be σ ε = ε0 1 − j (8.3.21) ωε0 5It is to be noted that electron has a different effective mass in a crystal lattice [71, 72], and hence, the electron mass has to be changed accordingly in the DLS model. 6This equation is similar to (8.3.18). In both cases, collision force dominates in the equation of motion (8.3.14). 88 Electromagnetic Field Theory Comparing (8.3.20) and (8.3.21), we see that ω 2 σ = ε p (8.3.22) 0 Γ The above formula for conductivity can be arrived at using collision frequency argument as is done in some textbooks [73]. As such, the DLS model is quite powerful: it can be used to explain a wide range of phenomena from very low frequency to optical frequency. The fact that ε < 0 can be used to explain many phenomena. The ionosphere is essentially a plasma medium described by ω 2 ε = ε 1 − p (8.3.23) 0 ω2 with ω0 = Γ = 0 called a cold collisionless plasma. Radio wave or microwave can only penetrate through this ionosphere when ω > ωp, so that ε > 0. The electrons in many conductive materials can be modeled as a sea of free electrons moving about quite freely with an effective mass . As such, they behave like a plasma medium as shall be seen. 8.3.5 Frequency Dispersive Media The DLS model shows that, except for vacuum, all media are frequency dispersive. It is prudent to digress to discuss more on the physical meaning of a frequency dispersive medium. The relationship between electric flux and electric field, in the frequency domain, still follows the formula D(ω) = ε(ω)E(ω) (8.3.24) When the effective permittivity, ε(ω), is a function of frequency, it implies that the above relationship in the time domain is via convolution, viz., D(t) = ε(t) ~ E(t) (8.3.25) Since the above represents a linear time-invariant (LTI) system [52], it implies that an input is not followed by an instantaneous output. In other words, there is a delay between the input and the output. The reason is because an electron has a mass, and it cannot respond immediately to an applied force: or it has inertial. (In other words, the system has memory of what it was before when you try to move it.) Even though the effective permittivity is a function of frequency, the frequency domain analysis we have done for a plane wave propagating in a dispersive medium still applies. For a mono-chromatic signal, it will have a velocity, called the phase velocity, given by √ v = 1/ µ0ε. Here, it also implies that different frequency components will propagate with different phase velocities through such a medium. Hence, a narrow pulse will spread in its width because different frequency components are not in phase after a short distance of travel. Also, the Lorentzian function is great for data fitting, as many experimentally observed resonances have finite Q and a line width. The Lorentzian function models that well. If multiple resonances occur in a medium or an atom, then multi-species DLS model can be Lossy Media, Lorentz Force Law, Drude-Lorentz-Sommerfeld Model 89 used. It is now clear that all media have to be frequency dispersive because of the finite mass of the electron and the inertial it has. In other words, there is no instantaneous response in a dielectric medium due to the finiteness of the electron mass. Even at optical frequency, many metals, which has a sea of freely moving electrons in the conduction band, can be modeled approximately as a plasma. A metal consists of a sea of electrons in the conduction band which are not tightly bound to the ions or the lattice. Also, in optics, the inertial force due to the finiteness of the electron mass (in this case effective mass , see Figure 8.4) can be sizeable compared to other forces. Then, ω0 ω or that the restoring force is much smaller than the inertial force, in (8.3.17), and if Γ is small, χ(ω) resembles that of a plasma, and ε of a metal can be negative. Figure 8.4: Effective masses of electron in different metals. 8.3.6 Plasmonic Nanoparticles When a plasmonic nanoparticle made of gold is excited by light, its response is given by (see homework assignment) 3 a cos θ εs − ε0 ΦR = E0 2 (8.3.26) r εs + 2ε0 In a plasma, εs can be negative, and thus, at certain frequency, if εs = −2ε0, then ΦR → ∞. Gold or silver with a sea of electrons, behaves like a plasma at optical frequencies, since the inertial force in the DLS model is quite large.7 Therefore, when light interacts with such a particle, it can sparkle brighter than normal. This reminds us of the saying “All that glitters is not gold!” even though this saying has a different intended meaning. Ancient Romans apparently knew about the potent effect of using gold and silver nanopar- ticles to enhance the reflection of light. These nanoparticles were impregnated in the glass or lacquer ware. By impregnating these particles in different media, the color of light will sparkle at different frequencies, and hence, the color of the glass emulsion can be changed (see website [74]). 7 2 2 2 In this case, ω ω0 , and ω ωΓ; the binding force and the collision force can be ignored similar to a cold plasma. 90 Electromagnetic Field Theory Figure 8.5: Ancient Roman goblets whose laquer coating glisten better due to the presence of gold nanoparticles. Gold or silver at optical frequencies behaves like plasma (courtesy of Smithsonian.com). Lecture 9 Waves in Gyrotropic Media, Polarization Gyrotropy is an important concept in electromagnetics. When a wave propagates through a gyrotropic medium, the electric field rotates changing the polarization of the wave. Our ionosphere is such a medium, and it affects radio and microwave communications between the Earth and the satellite by affecting the polarization of the wave. We will study this important topic in this lecture. 9.1 Gyrotropic Media and Faraday Rotation This section derives the effective permittivity tensor of a gyrotropic medium in the ionsphere. Our ionosphere is always biased by a static magnetic field due to the Earth’s magnetic field [75]. But in this derivation, to capture the salient feature of the physics with a simple model, we assume that the ionosphere has a static magnetic field polarized in the z direction, namely that B =zB ˆ 0. Now, the equation of motion from the Lorentz force law for an electron with q = −e, (in accordance with Newton’s second law that F = ma or force equals mass times accelration) becomes dv m = −e(E + v × B) (9.1.1) e dt Next, let us assume that the electric field is polarized in the xy plane. The derivative of v is the acceleration of the electron, and also, v = dr/dt where r =xx ˆ +yy ˆ +zz ˆ . Again, assuming linearity, we use frequency domain technique for the analysis. And in the frequency domain, the above equation in the cartesian coordinates becomes 2 meω x = e(Ex + jωB0y) (9.1.2) 2 meω y = e(Ey − jωB0x) (9.1.3) The above constitutes two equations with two unknowns x and y. They cannot be solved easily for x and y in terms of the electric field because they correspond to a two-by-two matrix system 91 92 Electromagnetic Field Theory with cross coupling between the unknowns x and y. But they can be simplified as follows: We can multiply (9.1.3) by ±j and add it to (9.1.2) to get two decoupled equations [76]: 2 meω (x + jy) = e[(Ex + jEy) + ωB0(x + jy)] (9.1.4) 2 meω (x − jy) = e[(Ex − jEy) − ωB0(x − jy)] (9.1.5) In the above, if we take the new unknowns to be x ± jy, the two equations are decoupled with respect to to these two unknowns. Defining new variables such that s± = x ± jy (9.1.6) E± = Ex ± jEy (9.1.7) then (9.1.4) and (9.1.5) become 2 meω s± = e(E± ± ωB0s±) (9.1.8) Thus, solving the above yields e s± = 2 E± = C±E± (9.1.9) meω ∓ eB0ω where e C± = 2 (9.1.10) meω ∓ eB0ω (By this manipulation, the above equations (9.1.2) and (9.1.3) transform to new equations where there is no cross coupling between s± and E±. The mathematical parlance for this is the diagonalization of a matrix equation [77]. Thus, the new equation can be solved easily.) Next, one can define Px = −Nex, Py = −Ney, and that P± = Px ± jPy = −Nes±. Then it can be shown that P± = ε0χ±E± (9.1.11) The expression for χ± can be derived, and they are given as 2 NeC± Ne e ωp χ± = − = − 2 = − 2 (9.1.12) ε0 ε0 meω ∓ eBoω ω ∓ Ωω 1 where Ω and ωp are the cyclotron frequency and plasma frequency, respectively. 2 eB0 2 Ne Ω = , ωp = (9.1.13) me meε0 At the cyclotron frequency, |χ±| → ∞. In other words, P± is finite even when E± = 0, or a solution exists to the equation of motion (9.1.1) without a forcing term, which in this case is the electric field. Thus, at this frequency, the solution blows up if the forcing term, E± is not 1This is also called the gyrofrequency. Waves in Gyrotropic Media, Polarization 93 zero. This is like what happens to an LC tank circuit at resonance whose current or voltage tends to infinity when the forcing term, like the voltage or current is nonzero. Now, one can express the original variables Px,Py,Ex,Ey in terms of P± and E±. With the help of (9.1.11), we arrive at P + P ε ε P = + − = 0 (χ E + χ E ) = 0 [χ (E + jE ) + χ (E − jE )] x 2 2 + + − − 2 + x y − x y ε = 0 [(χ + χ )E + j(χ − χ )E ] (9.1.14) 2 + − x + − y P − P ε ε P = + − = 0 (χ E − χ E ) = 0 [χ (E + jE ) − χ (E − jE )] y 2j 2j + + − − 2j + x y − x y ε = 0 [(χ − χ )E + j(χ + χ )E ] (9.1.15) 2j + − x + − y The above relationship in cartesian coordinates can be expressed using a tensor where P = ε0χ · E (9.1.16) where P = [Px,Py], and E = [Ex,Ey]. From the above, χ is of the form 2 2 ωp ωp Ω ! 1 (χ+ + χ−) j(χ+ − χ−) − ω2−Ω2 −j ω(ω2−Ω2) χ = = ω 2Ω ω 2 (9.1.17) 2 −j(χ+ − χ−)(χ+ + χ−) p p j ω(ω2−Ω2) − ω2−Ω2 Notice that in the above, when the B field is turned off or Ω = 0, the above resembles the solution of a collisionless, cold plasma again. For the B = 0 case with electric field in the x (or y) direction, it will drive a motion of the electron to be in the x (or y) direction. In this case, v × B term is zero, and the electron motion is unaffected by the magnetic field as can be seen from the Lorentz force law or (9.1.1). Hence, it behaves like a simple collisionless plasma without a biasing magnetic field. Consequently, for the B 6= 0 case, the above can be generalized to 3D to give χ0 jχ1 0 χ = −jχ1 χ0 0 (9.1.18) 0 0 χp 2 2 where χp = −ωp/ω . Notice that since we assume that B =zB ˆ 0, the z component of (9.1.1) is unaffected by the v × B force. Hence, the electron moving in the z is like that of a cold collisionless plasma. Using the fact that D = ε0E + P = ε0(I + χ) · E = ε · E, the above implies that 1 + χ0 jχ1 0 ε = ε0 −jχ1 1 + χ0 0 (9.1.19) 0 0 1 + χp Now, ε is that of an anisotropic medium, of which a gyrotropic medium belongs. Please notice that the above tensor is a hermitian tensor. We shall learn later that this is the hallmark of a lossless medium. 94 Electromagnetic Field Theory Another characteristic of a gyrotropic medium is that a linearly polarized wave will rotate when passing through it. This is the Faraday rotation effect [76], which we shall learn more later. This phenomenon poses a severe problem for Earth-to-satellite communication, using linearly polarized wave as it requires the alignment of the Earth-to-satellite antennas. This can be avoided using a rotatingly polarized wave, called a circularly polarized wave that we shall learn in the next section. As we have learnt, the ionosphere affects out communication systems two ways: It acts as a mirror for low-frequency electromagnetic or radio waves (making the experiment of Marconi a rousing success). It also affects the polarization of the wave. But the ionosphere of the Earth and the density of electrons that are ionized is highly dependent on temperature, and the effect of the Sun. The fluctuation of particles in the ionosphere gives rise to scintillation effects due to electron motion and collision that affect radio wave communication systems [78]. 9.2 Wave Polarization Studying wave polarization is very important for communication purposes [32]. A wave whose electric field is pointing in the x direction while propagating in the z direction is a linearly polarized (LP) wave. The same can be said of one with electric field polarized in the y direction. It turns out that a linearly polarized wave suffers from Faraday rotation when it propagates through the ionosphere. For instance, an x polarized wave can become a y polarized wave due to Faraday rotation. So its polarization becomes ambiguous as the wave propagates through the ionosphere: to overcome this, Earth to satellite communication is done with circularly polarized (CP) waves [79]. So even if the electric field vector is rotated by Faraday’s rotation, it remains to be a CP wave. We will study these polarized waves next. We can write a general uniform plane wave propagating in the z direction in the time domain as E =xE ˆ x(z, t) +yE ˆ y(z, t) (9.2.1) Clearly, ∇ · E = 0, and Ex(z, t) and Ey(z, t), by the principle of linear superposition, are so- lutions to the one-dimensional wave equation. For a time harmonic field, the two components may not be in phase, and we have in general Ex(z, t) = E1 cos(ωt − βz) (9.2.2) Ey(z, t) = E2 cos(ωt − βz + α) (9.2.3) where α denotes the phase difference between these two wave components. We shall study how the linear superposition of these two components behaves for different α’s. First, we set z = 0 to observe this field. Then E =xE ˆ 1 cos(ωt) +yE ˆ 2 cos(ωt + α) (9.2.4) π For α = 2 Ex = E1 cos(ωt), Ey = E2 cos(ωt + π/2) (9.2.5) Waves in Gyrotropic Media, Polarization 95 Next, we evaluate the above for different ωt’s ωt = 0, Ex = E1, Ey = 0 (9.2.6) √ √ ωt = π/4, Ex = E1/ 2, Ey = −E2/ 2 (9.2.7) ωt = π/2, Ex = 0, Ey = −E2 (9.2.8) √ √ ωt = 3π/4, Ex = −E1/ 2, Ey = −E2/ 2 (9.2.9) ωt = π, Ex = −E1, Ey = 0 (9.2.10) The tip of the vector field E traces out an ellipse as show in Figure 9.1. With the left-hand thumb pointing in the z direction, and the wave rotating in the direction of the fingers, such a wave is called left-hand elliptically polarized (LHEP) wave. Figure 9.1: If one follows the tip of the electric field vector, it traces out an ellipse as a function of time t. When E1 = E2, the ellipse becomes a circle, and we have a left-hand circularly polarized (LHCP) wave. When α = −π/2, the wave rotates in the counter-clockwise direction, and the wave is either right-hand elliptically polarized (RHEP), or right-hand circularly polarized (RHCP) wave depending on the ratio of E1/E2. Figure 9.2 shows the different polarizations of the wave wave for different phase differences and amplitude ratio. Figure 9.3 shows a graphic picture of a CP wave propagating through space. 96 Electromagnetic Field Theory Figure 9.2: Due to different phase difference between the Ex and Ey components of the field, and their relative amplitudes E2/E1, different polarizations will ensure. The arrow indicates the direction of rotation of the field vector. Figure 9.3: The rotation of the field vector of a right-hand circular polarization wave as it propagates in the right direction [80] (courtesy of Wikipedia). 9.2.1 Arbitrary Polarization Case and Axial Ratio2 As seen before, the tip of the field vector traces out an ellipse in space as it propagates. The axial ratio (AR) is the ratio of the major axis to the minor axis of this ellipse. It is an important figure of merit for designing CP (circularly polarized) antennas (antennas that will radiate circularly polarized waves). The closer is this ratio to 1, the better is the antenna design. We will discuss the general polarization and the axial ratio of a wave. For the general case for arbitrary α, we let Ex = E1 cos ωt, Ey = E2 cos(ωt + α) = E2(cos ωt cos α − sin ωt sin α) (9.2.11) 2This section is mathematically complicated. It can be skipped on first reading. Waves in Gyrotropic Media, Polarization 97 Then from the above, expressing Ey in terms of Ex, one gets " 2#1/2 E2 Ex Ey = Ex cos α − E2 1 − sin α (9.2.12) E1 E1 Rearranging and squaring, we get 2 2 aEx − bExEy + cEy = 1 (9.2.13) where 1 2 cos α 1 a = 2 2 , b = 2 , c = 2 2 (9.2.14) E1 sin α E1E2 sin α E2 sin α After letting Ex → x, and Ey → y, equation (9.2.13) is of the form, ax2 − bxy + cy2 = 1 (9.2.15) The equation of an ellipse in its self coordinates is x0 2 y0 2 + = 1 (9.2.16) A B where A and B are axes of the ellipse as shown in Figure 9.4. We can transform the above back to the (x, y) coordinates by letting x0 = x cos θ − y sin θ (9.2.17) y0 = x sin θ + y cos θ (9.2.18) to get cos2 θ sin2 θ 1 1 sin2 θ cos2 θ x2 + − xy sin 2θ − + y2 + = 1 (9.2.19) A2 B2 A2 B2 A2 B2 Comparing (9.2.13) and (9.2.19), one gets 1 −1 2 cos αE1E2 θ = tan 2 2 (9.2.20) 2 E2 − E1 1 + ∆1/2 AR = > 1 (9.2.21) 1 − ∆ where AR is the axial ratio and !1/2 4E 2E 2 sin2 α ∆ = 1 − 1 2 (9.2.22) 2 22 E1 + E2 98 Electromagnetic Field Theory Figure 9.4: This figure shows the parameters used to derive the axial ratio (AR) of an elliptically polarized wave. 9.3 Polarization and Power Flow For a linearly polarized wave in the time domain, E E =xE ˆ cos(ωt − βz), H =y ˆ 0 cos(ωt − βz) (9.3.1) 0 η Hence, the instantaneous power we have learnt previously in Section 5.3 becomes E 2 S(t) = E(t) × H(t) =z ˆ 0 cos2(ωt − βz) (9.3.2) η indicating that for a linearly polarized wave, the instantaneous power is function of both time and space. It travels as lumps of energy through space. In the above E0 is the amplitude of the linearly polarized wave. Next, we look at power flow for for elliptically and circularly polarized waves. It is to be noted that in the phasor world or frequency domain, (9.2.1) becomes −jβz −jβz+jα E(z, ω) =xE ˆ 1e +yE ˆ 2e (9.3.3) For LHEP wave, −jβz E(z, ω) = e (ˆxE1 + jyEˆ 2) (9.3.4) whereas for LHCP wave, −jβz E(z, ω) = e E1(ˆx + jyˆ) (9.3.5) Waves in Gyrotropic Media, Polarization 99 For RHEP wave, the above becomes −jβz E(z, ω) = e (ˆxE1 − jyEˆ 2) (9.3.6) whereas for RHCP wave, it is −jβz E(z, ω) = e E1(ˆx − jyˆ) (9.3.7) Focussing on the circularly polarized wave, −jβz E = (ˆx ± jyˆ)E0e (9.3.8) Using that β × E H = , ωµ where β =zβ ˆ , then E H = (∓xˆ − jyˆ)j 0 e−jβz (9.3.9) η where η = pµ/ε is the intrinsic impedance of the medium. Therefore, E(t) =xE ˆ 0 cos(ωt − βz) ± yEˆ 0 sin(ωt − βz) (9.3.10) E E H(t) = ∓xˆ 0 sin(ωt − βz) +y ˆ 0 cos(ωt − βz) (9.3.11) η η Then the instantaneous power becomes E 2 E 2 E 2 S(t) = E(t) × H(t) =z ˆ 0 cos2(ωt − βz) +z ˆ 0 sin2(ωt − βz) =z ˆ 0 (9.3.12) η η η In other words, a CP wave delivers constant instantaneous power independent of space and time, as opposed to a linearly polarized wave which delivers a non-constant instantaneous power as shown in (9.3.2). It is to be noted that the complex Poynting’s vector for a lossless medium S = E × H∗ (9.3.13) e is real and constant independent of space both for linearly, circularly, and elliptically polarized waves. This is because there is no reactive power in a plane wave of any polarization: the stored energy in the plane wave cannot be returned to the source! 100 Electromagnetic Field Theory Lecture 10 Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density Figure 10.1: The local coordinates used to describe a circularly polarized wave: In cartesian and polar coordinates. In the last lecture, we study circularly polarized waves as well as linearly polarized waves. In addition, these waves can carry power giving rise to power flow. But in addition to carrying power, a travelling wave also has a momentum: for a linearly polarized wave, it carries linear momention in the direction of the propagation of the traveling wave. But for a circularly polarized wave, it carries angular momentum as well. We have studied the complex Poynting’s theorem in the frequency domain with phasors. 101 102 Electromagnetic Field Theory Here, we will derive the lossless conditions for the permittivity and permeability tensors. Energy density is well defined for a lossless dispersionless medium, but it assumes a different formula when the medium is dispersive. 10.1 Spin Angular Momentum and Cylindrical Vector Beam In this section, we will study the spin angular momentum of a circularly polarized (CP) wave. It is to be noted that in cylindrical coordinates, as shown in Figure 10.1,x ˆ =ρ ˆcos φ − φˆ sin φ, yˆ =ρ ˆsin φ + φˆ cos φ, then a CP field is proportional to (ˆx ± jyˆ) =ρe ˆ ±jφ ± jφeˆ ±jφ = e±jφ(ˆρ ± φˆ) (10.1.1) Therefore, theρ ˆ and φˆ of a CP is also an azimuthal traveling wave in the φˆ direction in addition to being a traveling wave e−jβz in thez ˆ direction. This is obviated by writing e−jφ = e−jkφρφ (10.1.2) where kφ = 1/ρ is the azimuthal wave number, and ρφ is the arc length traversed by the azimuthal wave. Notice that the wavenumber kφ is dependent on ρ: the larger the ρ, the smaller is kφ, and hence, the larger the azimuthal wavelength. Thus, the wave possesses angular momentum called the spin angular momentum (SAM), just as a traveling wave e−jβz possesses linear angular momentum in thez ˆ direction. In optics research, the generation of cylindrical vector beam is in vogue. Figure 10.2 shows a method to generate such a beam. A CP light passes through a radial analyzer that will only allow the radial component of (10.1.1) to be transmitted. Then a spiral phase element (SPE) compensates for the exp(±jφ) phase shift in the azimuthal direction. Finally, the light is a cylindrical vector beam which is radially polarized without spin angular momentum. Such a beam has been found to have nice focussing property, and hence, has aroused researchers’ interest in the optics community [81]. Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density103 Figure 10.2: A cylindrical vector beam can be generated experimentally. The spiral phase element (SPE) compensates for the exp(±jφ) phase shift (courtesy of Zhan, Q. [81]). The half-wave plate rotates the polarization of a wave by 90 degrees. 10.2 Momentum Density of Electromagnetic Field We have seen that a traveling wave carries power and has energy density associated with it. In other words, the moving or traveling energy density gives rise to power flow. It turns out that a traveling wave also carries a momentum with it. The momentum density of electromagnetic field is given by G = D × B (10.2.1) also called the momentun density vector. With it, one can derive momentum conservation theorem [32, p. 59] [48]. The derivation is rather long, but we will justify the above formula and simplify the derivation using the particle or corpuscular nature of light or electromagnetic field. The following derivation is only valid for plane waves. It has been long known that electromagnetic energy is carried by photon, associated with a packet of energy given by ~ω. It is also well known that a photon has momentum given by ~k. Assuming that there are photons, with density of N photons per unit volume, streaming through space at the velocity of light c. Then the power flow associated with these streaming photons is given by E × H = ~ωNczˆ (10.2.2) 104 Electromagnetic Field Theory Assuming that the plane wave is propagating in the z direction. Using k = ω/c, we can rewrite the above more suggestively as 2 E × H = ~ωNczˆ = ~kNc zˆ (10.2.3) where k = ω/c. Defining the momentum density vector to be G = ~kNzˆ (10.2.4) From the above, we deduce that 1 E × H = Gc2 = G (10.2.5) µε Or the above can be rewritten as G = D × B (10.2.6) where D = εE, and B = µH.1 10.3 Complex Poynting’s Theorem and Lossless Condi- tions 10.3.1 Complex Poynting’s Theorem It has been previously shown that the vector E(r, t) × H(r, t) has a dimension of watts/m2 which is that of power density. Therefore, it is associated with the direction of power flow [32, 48]. As has been shown for time-harmonic field, a time average of this vector can be defined as ¢T 1 hE(r, t) × H(r, t)i = lim E(r, t) × H(r, t) dt. (10.3.1) T →∞ T 0 Given the phasors of time harmonic fields E(r, t) and H(r, t), namely, E(r, ω) and H(r, ω) respectively, we can show that 1 hE(r, t) × H(r, t)i = S(r, t) = E(r, t) × H(r, t) (10.3.3) and the complex Poynting’s vector to be S(r, ω) = E(r, ω) × H∗(r, ω) (10.3.4) ^ 1The author is indebted to Wei SHA for this simple derivation. Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density105 then for time-harmonic fields, 1 hS(r, t)i = 10.3.2 Lossless Conditions For a region V that is lossless and source-free, J = 0. There should be no net time-averaged power-flow out of or into this region V . Therefore, ¢ S Because of energy conservation, the real part of the right-hand side of (10.3.11), without the E · J∗ term, must be zero. In other words, the right-hand side of (10.3.11) should be purely imaginary. Thus ¢ dV (H∗ · µ · H − E · ε∗ · E∗) (10.3.14) V 2We will drop the argument r, ω for the phasors in our discussion next as they will be implied. 106 Electromagnetic Field Theory must be a real quantity. Other than the possibility that the above is zero, the general requirement for (10.3.14) to be real for arbitrary E and H, is that H∗ · µ · H and E · ε∗ · E∗ are real quantities. This is only possible if µ is hermitian.3 Therefore, the conditions for anisotropic media to be lossless are µ = µ†, ε = ε†, (10.3.15) requiring the permittivity and permeability tensors to be hermitian. If this is the case, (10.3.14) is always real for arbitraty E and H, and (10.3.13) is true, implying a lossless region V . Notice that for an isotropic medium, this lossless conditions reduce simply to that =m(µ) = 0 and =m(ε) = 0, or that µ and ε are pure real quantities. Looking back, many of the effective permittivities or dielectric constants that we have derived using the Drude-Lorentz-Sommerfeld model cannot be lossless when the friction term is involved. For a lossy medium which is conductive, we may define J = σ · E where σ is a general conductivity tensor. In this case, equation (10.3.12), after combining the last two terms, may be written as ¢ ¢ jσ∗ dS · (E × H∗) = −jω dV H∗ · µ · H − E · ε∗ + · E∗ (10.3.16) ω S V¢ ∗ = −jω dV [H∗ · µ · H − E · ε˜ · E∗], (10.3.17) ˜ jσ where ε = ε − ω which is the general complex permittivity tensor. In this manner, (10.3.17) has the same structure as the source-free Poynting’s theorem. Notice here that the complex permittivity tensor ε˜ is clearly non-hermitian corresponding to a lossy medium. For a lossless medium without the source term, by taking the imaginary part of (10.3.12), we arrive at ¢ ¢ =m dS · (E × H∗) = −ω dV (H∗ · µ · H − E · ε∗ · E∗), (10.3.18) S V The left-hand side of the above is the reactive power coming out of the volume V , and hence, the right-hand side can be interpreted as reactive power as well. It is to be noted that H∗ · µ · H and E · ε∗ · E∗ are not to be interpreted as stored energy density when the medium is dispersive. The correct expressions for stored energy density will be derived in the next section. But, the quantity H∗ · µ · H for lossless, dispersionless media is associated with the time- averaged energy density stored in the magnetic field, while the quantity E · ε∗ · E∗ for lossless dispersionless media is associated with the time-averaged energy density stored in the electric field. Then, for lossless, dispersionless, source-free media, then the right-hand side of the 3H∗ · µ · H is real only if its complex conjugate, or conjugate transpose is itself. Using some details from t t t matrix algebra that (A · B · C)t = C · B · A , implies that (in physics notation, the transpose of a vector is implied in a dot product) (H∗ · µ · H)† = (H · µ∗ · H∗)t = H∗ · µ† · H = H∗ · µ · H. The last equality in the above is possible only if µ = µ† or that µ is hermitian. Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density107 above can be interpreted as stored energy density. Hence, the reactive power is proportional to the time rate of change of the difference of the time-averaged energy stored in the magnetic field and the electric field. 10.4 Energy Density in Dispersive Media A dispersive medium alters our concept of what energy density is.4 To this end, we assume that the field has complex ω dependence in ejωt, where ω = ω0 − jω00, rather than real ω de- pendence. We take the divergence of the complex power for fields with such time dependence, and let ejωt be attached to the field. So E(t) and H(t) are complex field but not exactly like phasors since they are not truly time harmonic. In other words, we let E(r, t) = E(r, ω)ejωt, H(r, t) = H(r, ω)ejωt (10.4.1) e e The above, just like phasors, can be made to satisfy Maxwell’s equations where the time derivative becomes jω but with complex ω. We can study the quantity E(r, t) × H∗(r, t) which has the unit of power density. In the real ω case, their time dependence will exactly cancel each other and this quantity becomes complex power again, but not in the complex ω case. Hence, ∇ · [E(t) × H∗(t)] = H∗(t) · ∇ × E(t) − E(t) · ∇ × H∗(t) = −H∗(t) · jωµH(t) + E(t) · jω∗ε∗E∗(t) (10.4.2) where Maxwell’s equations have been used to substitute for ∇ × E(t) and ∇ × H∗(t). The space dependence of the field is implied, and we assume a source-free medium so that J = 0. ∗ If E(t) ∼ ejωt, then, due to ω being complex, now H∗(t) ∼ e−jω t, and the term like E(t) × H∗(t) is not truly time independent but becomes ∗ 00 E(t) × H∗(t) ∼ ej(ω−ω )t = e2ω t (10.4.3) And each of the term above will have similar time dependence. Writing (10.4.2) more explic- itly, by letting ω = ω0 − jω00, we have ∇ · [E(t) × H∗(t)] = −j(ω0 − jω00)µ(ω)|H(t)|2 + j(ω0 + jω00)ε∗(ω)|E(t)|2 (10.4.4) Assuming that ω00 ω0, or that the field is quasi-time-harmonic, we can let, after using Taylor series approximation, that ∂µ(ω0) ∂ε(ω0) µ(ω0 − jω00) =∼ µ(ω0) − jω00 , ε(ω0 − jω00) =∼ ε(ω0) − jω00 (10.4.5) ∂ω0 ∂ω0 4The derivation here is inspired by H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements [82]. Generalization to anisotropic media is given by W.C. Chew, Lectures on Theory of Microwave and Optical Waveguides [83]. 108 Electromagnetic Field Theory Using (10.4.5) in (10.4.4), and collecting terms of the same order, and ignoring (ω00)2 terms, gives5 ∇ · [E(t) × H∗(t)] =∼ − jω0µ(ω0)|H(t)|2 + jω0ε∗(ω0)|E(t)|2 ∂µ − ω00µ(ω0)|H(t)|2 − ω0ω00 |H(t)|2 ∂ω0 ∂ε∗ − ω00ε∗(ω0)|E(t)|2 − ω0ω00 |E(t)|2 (10.4.6) ∂ω0 The above can be rewritten as ∇ · [E(t) × H∗(t)] =∼ −jω0 µ(ω0)|H(t)|2 − ε∗(ω0)|E(t)|2 ∂ω0µ(ω0) ∂ω0ε∗(ω0) − ω00 |H(t)|2 + |E(t)|2 (10.4.7) ∂ω0 ∂ω0 The above approximation is extremely good when ω00 ω0. For a lossless medium, ε(ω0) and µ(ω0) are purely real, and the first term of the right-hand side is purely imaginary while the second term is purely real. In the limit when ω00 → 0, when half the imaginary part of the above equation is taken, we have 1 1 1 ∇ · =m [E × H∗] = −ω0 µ|H|2 − ε|E|2 (10.4.8) 2 2 2 which has the physical interpretation of reactive power as has been previously discussed. When half the real part of (10.4.7) is taken, we obtain 1 ω00 ∂ω0µ ∂ω0ε ∇ · 00 Since the right-hand side has time dependence of e2ω t, it can be written as 1 ∂ 1 ∂ω0µ ∂ω0ε ∂ ∇ · 1 ∂ω0µ ∂ω0ε hW i = |H|2 + |E|2 (10.4.11) T 4 ∂ω0 ∂ω0 For a non-dispersive medium, the above reverts back to a familiar expression, 1 hW i = µ|H|2 + ε|E|2 (10.4.12) T 4 which is what we have derived before. In the above analysis, we have used a quasi-time-harmonic signal with exp(jωt) depen- dence. In the limit when ω00 → 0, this signal reverts back to a time-harmonic signal, and to 5This is the general technique of perturbation expansion [45]. Spin Angular Momentum, Complex Poynting’s Theorem, Lossless Condition, Energy Density109 our usual interpretation of complex power. However, by assuming the frequency ω to have a small imaginary part ω00, it forces the stored energy to grow very slightly, and hence, power has to be supplied to maintain the growth of this stored energy. By so doing, it allows us to identify the expression for energy density for a dispersive medium. These expressions for energy density were not discovered until 1960 by Brillouin [84], as energy density times group velocity should be power flow. More discussion on this topic can be found in Jackson [48]. It is to be noted that if the same analysis is used to study the energy storage in a capacitor or an inductor, the energy storage formulas have to be accordingly modified if the capacitor or inductor is frequency dependent. 110 Electromagnetic Field Theory Lecture 11 Transmission Lines Transmission lines represent one of the most important electromagnetic technologies. The reason being that they can be described by simple theory, similar to circuit theory. As such, the theory is within the grasp of most practicing electrical engineers. Moreover, transmission line theory fills the gap in the physics of circuit theory: Circuit theory alone cannot describe wave phenomena, but when circuit theory is augmented with transmission line theory, wave phenomena with its corresponding wave physics start to emerge. Even though circuit theory has played an indispensable role in the development of the computer chip industry, it has to be embellished by transmission line theory, so that high- speed circuits can be designed. Retardation effect, which causes time delay, clock skew, and phase shift, can be modeled simply using transmission lines. Nowadays, commercial circuit solver software have the capability of including transmission line as an element in modeling. 111 112 Electromagnetic Field Theory 11.1 Transmission Line Theory Figure 11.1: Various kinds of transmission lines. Schematically, all of them can be modeled by two parallel wires. Transmission lines were the first electromagnetic waveguides ever invented. The were driven by the needs in telegraphy technology. It is best to introduce transmission line theory from the viewpoint of circuit theory, which is elegant and one of the simplest theories of electrical engineering. This theory is also discussed in many textbooks and lecture notes. Transmission lines are so important in modern day electromagnetic engineering, that most engineering electromagnetics textbooks would be incomplete without introducing the topics related to them [31, 32,44, 50,54, 65,79,83, 85,86]. Circuit theory is robust and is not sensitive to the detail shapes of the components in- volved such as capacitors or inductors. Moreover, many transmission line problems cannot be analyzed simply when the full form of Maxwell’s equations is used,1 but approximate solutions can be obtained using circuit theory in the long-wavelength limit. We shall show that circuit theory is an approximation of electromagnetic field theory when the wavelength is very long: the longer the wavelength, the better is the approximation [50]. Examples of transmission lines are shown in Figure 11.1. The symbol for a transmission line is usually represented by two pieces of parallel wires, but in practice, these wires need not be parallel as showin in Figure 11.2. 1Usually called full-wave analysis. Transmission Lines 113 Figure 11.2: A twisted pair transmission line where the two wires are not parallel to each other (courtesy of slides by A. Wadhwa, A.L. Dal, N. Malhotra [87].) Circuit theory also explains why waveguides can be made sloppily when wavelength is long or the frequency low. For instance, in the long-wavelength limit, we can make twisted-pair waveguides with abandon, and they still work well (see Figure 11.2). Hence, it is simplest to first explain the propagation of electromagnetic signal on a transmission line using circuit analysis. 11.1.1 Time-Domain Analysis We will start with performing the time-domain analysis of a simple, infinitely long transmis- sion line. Remember that two pieces of metal can accumulate attractive positive and negative charges between them, giving rise to capacitive coupling, with electric fields that start with positive charges and end with negative charges, and hence stored energy in the electric field. Moreover, a piece of wire carrying a current generates a magnetic field, and hence, yields stored energy in the magnetic field. These stored energies are the sources of the capacitive and inductive effects. But these capacitive and inductive effects are distributed over the spa- tial dimension of the transmission line. Therefore, it is helpful to think of the two pieces of metal as consisting of small segments of metal concatenated together. Each of these segments will have a small inductance, as well as a small capacitive coupling between them. Hence, we can model two pieces of metal with a distributed lumped element model as shown in Figure 11.3. For simplicity, we assume the other conductor to be a ground plane, so that it need not be approximated with lumped elements. In the transmission line, the voltage V (z, t) and the current I(z, t) are functions of both space z and time t, but we will model the space variation of the voltage and current with discrete step approximations. The voltage varies from node to node while the current varies from branch to branch of the lump-element model. 114 Electromagnetic Field Theory Figure 11.3: A long transmission line can be replaced by a concatenation of many short transmission lines. For each pair of short wires, there are capacitive coupling between them. Furtheremore, when current flows in the wire, magnetic field is generated making them behave like an inductor. Therefore, it can be replaced by a lumped-element approximation as shown. Telegrapher’s Equations First, we recall that the V-I relation of an inductor is dI V = L 0 (11.1.1) 0 0 dt where L0 is the inductor, V0 is the time-varying voltage drop across the inductor, and I0 is the current through the inductor. Then using this relation between nodes 1 and 2, we have ∂I V − (V + ∆V ) = L∆z (11.1.2) ∂t The left-hand side is the voltage drop across the inductor, while the right-hand side follows from the aforementioned V-I relation of an inductor, but we have replaced L0 = L∆z. Here, L is the inductance per unit length (line inductance) of the transmission line. And L∆z is the incremental inductance due to the small segment of metal of length ∆z. In the above, we assume that V = V (z, t) and I = I(z, t), so that time derivative is replaced by partial time derivative. Then the above can be simplified to ∂I ∆V = −L∆z (11.1.3) ∂t Next, we make use of the V-I relation for a capacitor, which is dV I = C 0 (11.1.4) 0 0 dt where C0 is the capacitor, I0 is the current through the capacitor, and V0 is a time-varying voltage drop across the capacitor. Thus, applying this relation at node 2 gives ∂ ∂V −∆I = C∆z (V + ∆V ) ≈ C∆z (11.1.5) ∂t ∂t Transmission Lines 115 where C is the capacitance per unit length, and C∆z is the incremental capacitance between the small piece of metal and the ground plane. In the above, we have used Kirchhoff current law to surmise that the current through the capacitor is −∆I, where ∆I = I(z+∆z, t)−I(z, t). In the last approximation in (11.1.5), we have dropped a term involving the product of ∆z and ∆V , since it will be very small or second order in magnitude. In the limit when ∆z → 0, one gets from (11.1.3) and (11.1.5) that ∂V (z, t) ∂I(z, t) = −L (11.1.6) ∂z ∂t ∂I(z, t) ∂V (z, t) = −C (11.1.7) ∂z ∂t The above are the telegrapher’s equations.2 They are two coupled first-order equations, and can be converted into second-order equations easily. Therefore, ∂2V ∂2V − LC = 0 (11.1.8) ∂z2 ∂t2 ∂2I ∂2I − LC = 0 (11.1.9) ∂z2 ∂t2 The above are wave equations that we have previously studied, where the velocity of the wave is given by 1 v = √ (11.1.10) LC Furthermore, if we assume that V (z, t) = V0f+(z − vt),I(z, t) = I0f+(z − vt) (11.1.11) corresponding to a right-traveling wave, they can be verified to satisfy (11.1.6) and (11.1.7) by back substitution. Consequently, we can easily show that r V (z, t) L = = Z (11.1.12) I(z, t) C 0 where Z0 is the characteristic impedance of the transmission line. The above ratio is only true for one-way traveling wave, in this case, one that propagates in the +z direction. For a wave that travels in the negative z direction, we can let, V (z, t) = V0f−(z + vt),I(z, t) = I0f−(z + vt) (11.1.13) one can easily show that r V (z, t) L = − = −Z (11.1.14) I(z, t) C 0 2They can be thought of as the distillation of the Faraday’s law and Ampere’s law from Maxwell’s equations without the source term. Their simplicity gives them an important role in engineering electromagnetics. 116 Electromagnetic Field Theory Time-domain analysis is very useful for transient analysis of transmission lines, especially when nonlinear elements are coupled to the transmission line.3 Another major strength of transmission line model is that it is a simple way to introduce time-delay (also called retardation) in a simple circuit model.4 Time delay is a wave propagation effect, and it is harder to incorporate into circuit theory or a pure circuit model consisting of R, L, and C only. In circuit theory, where the wavelength is assumed very long, Laplace’s equation is usually solved, which is equivalent to Helmholtz equation with infinite wave velocity, namely, ω2 lim ∇2Φ(r) + Φ(r) = 0 =⇒ ∇2Φ(r) = 0 (11.1.15) c→∞ c2 Hence, events in Laplace’s equation happen instantaneously. In other words, circuit theory assumes that the velocity of the wave is infinite, and there is no retardation effect. This is only true or a good approximation when the size of the structure is small compared to wavelength. 11.1.2 Frequency-Domain Analysis–the Power of Phasor Technique Again! Frequency domain analysis is very popular as it makes the transmission line equations very simple–one just replace ∂/∂t → jω. Moreover, generalization to a lossy system is quite straight forward. Furthermore, for linear time invariant systems, the time-domain signals can be obtained from the frequency-domain data by performing a Fourier inverse transform since phasors and Fourier transforms of a time variable are just related to each other by a constant. The telegrapher’s equations (11.1.6) and (11.1.7) then in frequency domain become d V (z, ω) = −jωLI(z, ω) (11.1.16) dz d I(z, ω) = −jωCV (z, ω) (11.1.17) dz The corresponding Helmholtz equations are then d2V + ω2LCV = 0 (11.1.18) dz2 d2I + ω2LCI = 0 (11.1.19) dz2 The above are second ordinary differential equations, and the general solutions to the above are −jβz jβz V (z) = V+e + V−e (11.1.20) −jβz jβz I(z) = I+e + I−e (11.1.21) 3Remember that we can only use frequency domain technique or Fourier transform for linear time-invariant systems. 4By a simple circuit model, we mean a model that has lumped elements such as R, L, and C as well as a transmission line element. Transmission Lines 117 √ where β = ω LC. This is similar to what we have seen previously for plane waves in the one-dimensional wave equation in free space, where −jk0z jk0z Ex(z) = E0+e + E0−e (11.1.22) √ where k0 = ω µ00. We see much similarity between (11.1.20), (11.1.21), and (11.1.22). jφ± To see the solution in the time domain, we let the phasor V± = |V±|e in (11.1.20), and the voltage signal above can then be converted back to the time domain using the key formula in phasor technique as V (z, t) = = |V+| cos(ωt − βz + φ+) + |V−| cos(ωt + βz + φ−) (11.1.24) As can be seen, the first term corresponds to a right-traveling wave, while the second term is a left-traveling wave. Furthermore, if we assume only a one-way traveling wave to the right by letting V− = I− = 0, then it can be shown that, for a right-traveling wave, using (11.1.16) or (11.1.17) r V (z) V+ L = = = Z0 (11.1.25) I(z) I+ C where Z0 is the characteristic impedance. Since Z0 is real, it implies that V (z) and I(z) are in phase. Similarly, applying the same process for a left-traveling wave only, by letting V+ = I+ = 0, then r V (z) V− L = = − = −Z0 (11.1.26) I(z) I− C In other words, for the left-traveling waves, the voltage and current are 180 ◦ out of phase. 11.2 Lossy Transmission Line Figure 11.4: In a lossy transmission line, series resistance can be added to the series induc- tance, and the shunt conductance can be added to the shun susceptance of the capacitor. However, this problem is homomorphic to the lossless case in the frequency domain. 118 Electromagnetic Field Theory The power of frequency domain analysis is revealed in the study of lossy transmission lines. The previous analysis, which is valid for lossless transmission line, can be easily generalized to the lossy case in the frequency domain. In using frequency domain and phasor technique, impedances will become complex numbers as shall be shown. To include loss, we use the lumped-element model as shown in Figure 11.4. One thing to note is that jωL is actually the series line impedance of the lossless transmission line, while jωC is the shunt line admittance of the same line. First, we can rewrite the expressions for the telegrapher’s equations in (11.1.16) and (11.1.17) in terms of series line impedance and shunt line admittance to arrive at d V = −ZI (11.2.1) dz d I = −YV (11.2.2) dz where Z = jωL and Y = jωC. The above can be easily generalized to the lossy case as shall be shown. The geometry in Figure 11.4 is topologically similar to, or homomorphic5 to the lossless case in Figure 11.3. Hence, when lossy elements are added in the geometry, we can surmise that the corresponding telegrapher’s equations are similar to those above. But to include loss, we need only to generalize the series line impedance and shunt admittance from the lossless case to lossy case as follows: Z = jωL → Z = jωL + R (11.2.3) Y = jωC → Y = jωC + G (11.2.4) where R is the series line resistance, and G is the shunt line conductance, and now Z and Y are the series impedance and shunt admittance, (which are complex number rather than being pure imaginary), respectively. Then, the corresponding Helmholtz equations are d2V − ZYV = 0 (11.2.5) dz2 d2I − ZYI = 0 (11.2.6) dz2 or d2V − γ2V = 0 (11.2.7) dz2 d2I − γ2I = 0 (11.2.8) dz2 where γ2 = ZY , or that one can also think of γ2 = −β2 by comparing with (11.1.18) and (11.1.19). Then the above is homomorphic to the lossless case except that now, β is a complex 5A math term for “similar in math structure”. The term is even used in computer science describing a emerging field of homomorphic computing. Transmission Lines 119 number, indicating that the field is decaying as it propagates. As before, the above are second order one-dimensional Helmholtz equations where the general solutions are −γz γz V (z) = V+e + V−e (11.2.9) −γz γz I(z) = I+e + I−e (11.2.10) and √ γ = ZY = p(jωL + R)(jωC + G) = jβ (11.2.11) Here, β = β0 − jβ00 is now a complex number. In other words, 0 00 e−γz = e−jβ z−β z is an oscillatory and decaying wave. Or focusing on the voltage case, −β00z−jβ0z β00z+jβ0z V (z) = V+e + V−e (11.2.12) jφ± Again, letting V± = |V±|e , the above can be converted back to the time domain as V (z, t) = The first term corresponds to a decaying wave moving to the right while the second term is also a decaying wave but moving to the left. When there is no loss, or√R = G = 0, and from (11.2.11), we retrieve the lossless case where β00 = 0 and γ = jβ = jω LC. Notice that for the lossy case, the characteristic impedance, which is the ratio of the voltage to the current for a one-way wave, can similarly be derived using homomorphism: r s r s V+ V− L jωL Z jωL + R Z0 = = − = = → Z0 = = (11.2.15) I+ I− C jωC Y jωC + G The above Z0 is manifestly a complex number. Here, Z0 is the ratio of the phasors of the one-way traveling waves, and apparently, their current phasor and the voltage phasor will not be in phase for lossy transmission line. In the absence of loss, the above again becomes r L Z = (11.2.16) 0 C the characteristic impedance for the lossless case previously derived. 120 Electromagnetic Field Theory Lecture 12 More on Transmission Lines As mentioned before, transmission line theory is indispensable in microwave engineering these days. The theory is the necessary augmentation of circuit theory for higher frequency analysis, and it is also indispensable to integrated circuit designers as computer clock rate becomes faster. Over the years, engineers have developed some very useful tools and measurement techniques to expand the design space of circuit designers. We will learn some of these tools in this lecture.1 As seen in the previous lecture, the telegrapher’s equations are similar to the one-dimensional form of Maxwell’s equations, and can be thought of as Maxwell’s equations in their simplest form. Therefore, they entail a subset of the physics seen in the full Maxwell’s equations. 12.1 Terminated Transmission Lines Figure 12.1: A schematic for a transmission line terminated with an impedance load ZL at z = 0. For an infinitely long transmission line, the solution consists of the linear superposition of a wave traveling to the right plus a wave traveling to the left. If transmission line is terminated 1Some of you may have studied this topic in your undergraduate electromagnetics course. However, this topic is important, and you will have to muster your energy to master this knowledge again:) 121 122 Electromagnetic Field Theory by a load as shown in Figure 12.1, a right-traveling wave will be reflected by the load, and in general, the wave on the transmission line will be a linear superposition of the left and right traveling waves. To simplify the analysis, we will assume that the line is lossless. The generalization to the lossy case is quite straightforward. Thus, we assume that −jβz jβz V (z) = a+e + a−e = V+(z) + V−(z) (12.1.1) √ where β = ω LC. In the above, in general, a+ 6= a−. Besides, this is a linear system; hence, we can define the right-going wave V+(z) to be the input, and that the left-going wave V−(z) to be the output as due to the reflection of the right-going wave V+(z). Or we can define the amplitude of the left-going reflected wave a− to be linearly related to the amplitude of the right-going or incident wave a+. In other words, at z = 0, we can let V−(z = 0) = ΓLV+(z = 0) (12.1.2) thus, using the definition of V+(z) and V−(z) as implied in (12.1.1), we have a− = ΓLa+ (12.1.3) where ΓL is the termed the reflection coefficient. Hence, (12.1.1) becomes −jβz jβz −jβz jβz V (z) = a+e + ΓLa+e = a+ e + ΓLe (12.1.4) The corresponding current I(z) on the transmission line is given by using the telegrapher’s equations as previously defined. By recalling that dV = −jωLI dz then for the general case, a+ −jβz jβz I(z) = e − ΓLe (12.1.5) Z0 Notice the sign change in the second term of the above expression. Similar to ΓL, a general reflection coefficient can be defined (which is a function of z) relating the left-traveling and right-traveling wave at location z such that jβz jβz V−(z) = a−e a−e 2jβz Γ(z) = −jβz = −jβz = ΓLe (12.1.6) V+(z) = a+e a+e Of course, Γ(z = 0) = ΓL. Furthermore, due to the V-I relation at an impedance load, we must have2 V (z = 0) = Z (12.1.7) I(z = 0) L 2One can also look at this from a differential equation viewpoint that this is a boundary condition. More on Transmission Lines 123 or that using (12.1.4) and (12.1.5) with z = 0, the left-hand side of the above can be rewritten, and we have 1 + ΓL 1 + ΓL ZL Z0 = ZL, or = = ZnL (12.1.8) 1 − ΓL 1 − ΓL Z0 From the above, we can solve for ΓL in terms of ZnL to get ZnL − 1 ZL − Z0 ΓL = = (12.1.9) ZnL + 1 ZL + Z0 Thus, given the termination load ZL and the characteristic impednace Z0, the reflection coefficient ΓL can be found, or vice versa. Or given ΓL, the normalized load impedance, ZnL = ZL/Z0, can be found. It is seen that ΓL = 0 if ZL = Z0. Thus a right-traveling wave will not be reflected and the left-traveling is absent. This is the case of a matched load. When there is no reflection, all energy of the right-traveling wave will be totally absorbed by the load. In general, we can define a generalized impedance at z 6= 0 to be V (z) a (e−jβz + Γ ejβz) Z(z) = = + L 1 −jβz jβz I(z) a+(e − ΓLe ) Z0 2jβz 1 + ΓLe 1 + Γ(z) = Z0 2jβz = Z0 (12.1.10) 1 − ΓLe 1 − Γ(z) where Γ(z) defined in (12.1.6) is used. The above can also be written as 1 + Γ(z) Z (z) = Z(z)/Z = (12.1.11) n 0 1 − Γ(z) where Zn(z) is the normalized generalized impedance. Conversely, one can write the above as Z (z) − 1 Z(z) − Z Γ(z) = n = 0 (12.1.12) Zn(z) + 1 Z(z) + Z0 From (12.1.10) above, one gets 2jβz 1 + ΓLe Z(z) = Z0 2jβz (12.1.13) 1 − ΓLe One can show that by setting z = −l, using (12.1.9), and after some algebra, ZL + jZ0 tan βl Z(−l) = Z0 (12.1.14) Z0 + jZL tan βl 124 Electromagnetic Field Theory 12.1.1 Shorted Terminations Figure 12.2: The input reactance (X) of a shorted transmission line as a function of its length l. From (12.1.14) above, when we have a short such that ZL = 0, then Z(−l) = jZ0 tan(βl) = jX (12.1.15) When βl 1, then tan βl ≈ βl, and (12.1.15) becomes ∼ Z(−l) = jZ0βl (12.1.16) p √ After using that Z0 = L/C and that β = ω LC, (12.1.16) becomes Z(−l) =∼ jωLl (12.1.17) The above implies that a short length of transmission line connected to a short as a load looks like an inductor with Leff = Ll, since much current will pass through this short producing a strong magnetic field with stored magnetic energy. Remember here that L is the line inductance, or inductance per unit length. On the other hand, when the length of the shorted line increases, due to the standing wave on the transmission line, certainly parts of the line will have charge accumulation giving rise to strong electric field, while other parts have current flow giving rise to strong magnetic field. Depending on this standing wave pattern, the line can become either capacitive or inductive. More on Transmission Lines 125 12.1.2 Open Terminations Figure 12.3: The input reactance (X) of an open transmission line as a function of its length l. When we have an open circuit such that ZL = ∞, then from (12.1.14) above Z(−l) = −jZ0 cot(βl) = jX (12.1.18) Then, when βl 1, cot(βl) ≈ 1/βl Z Z(−l) ≈ −j 0 (12.1.19) βl √ p And then, again using β = ω LC, Z0 = L/C 1 Z(−l) ≈ (12.1.20) jωCl Hence, an open-circuited terminated short length of transmission line appears like an effective capacitor with Ceff = Cl. Again, remember here that C is line capacitance or capacitance per unit length. Again, as shown in Figure 12.3, the impedance at z = −l is purely reactive, and goes through positive and negative values due to the standing wave set up on the transmission line. Therefore, by changing the length of l, one can make a shorted or an open terminated line look like an inductor or a capacitor depending on its length l. This effect is shown in Figures 12.2 and 12.3. Moreover, the reactance X becomes infinite or zero with the proper choice of the length l. These are resonances or anti-resonances of the transmission line, very much like an LC tank circuit. An LC circuit can look like an open or a short circuit at resonances and depending on if they are connected in parallel or in series. 126 Electromagnetic Field Theory 12.2 Smith Chart In general, from (12.1.13) and (12.1.14), a length of transmission line can transform a load ZL to a range of possible complex values Z(−l). To understand this range of values better, we can use the Smith chart (invented by P.H. Smith 1939 before the advent of the computer) [88]. The Smith chart is essentially a graphical calculator for solving transmission line problems. It has been used so much by microwave engineers during the early days that its use has imposed a strong impression on these engineers: it also has become an indispensable visual aid for understanding and solving microwave engineering problems. Equation (12.1.12) indicates that there is a unique map between the normalized impedance Zn(z) = Z(z)/Z0 and reflection coefficient Γ(z). In the normalized impedance form where Zn = Z/Z0, from (12.1.10) and (12.1.12) Zn − 1 1 + Γ Γ = ,Zn = (12.2.1) Zn + 1 1 − Γ Equations in (12.2.1) are related to a bilinear transform in complex variables [89]. It is a conformal map that maps circles to circles. Such a map is shown in Figure 12.4, where lines on the right-half of the complex Zn plane are mapped to the circles on the complex Γ plane. Since straight lines on the complex Zn plane are circles with infinite radii, they are mapped to circles on the complex Γ plane. The Smith chart shown on Figure 12.5 allows one to obtain the corresponding Γ given Zn and vice versa as indicated in (12.2.1), but using a graphical calculator or the Smith chart. Notice that the imaginary axis on the complex Zn plane maps to the circle of unit radius on the complex Γ plane. All points on the right-half plane are mapped to within the unit circle. The reason being that the right-half plane of the complex Zn plane corresponds to passive impedances that will absorb energy. Hence, such an impedance load will have reflection coefficient with amplitude less than one, which are points within the unit circle. On the other hand, the left-half of the complex Zn plane corresponds to impedances with negative resistances. These will be active elements that can generate energy, and hence, yielding |Γ| > 1, and will be outside the unit circle on the complex Γ plane. Another point to note is that points at infinity on the complex Zn plane map to the point at Γ = 1 on the complex Γ plane, while the point zero on the complex Zn plane maps to Γ = −1 on the complex Γ plane. These are the reflection coefficients of an open-circuit load and a short-circuit load, respectively. For a matched load, Zn = 1, and it maps to the zero point on the complex Γ plane implying no reflection. More on Transmission Lines 127 Zn−1 1+Γ Figure 12.4: Bilinear map of the formulae Γ = , and Zn = . The chart on the right, Zn+1 1−Γ called the Smith chart, allows the values of Zn to be determined quickly given Γ, and vice versa. The Smith chart also allows one to quickly evaluate the expression −2jβl Γ(−l) = ΓLe (12.2.2) and its corresponding Zn not by using (12.2.1) via a calculator, but by using a graphical calculator—the Smith chart. Since β = 2π/λ, it is more convenient to write βl = 2πl/λ, and measure the length of the transmission line in terms of wavelength. To this end, the above becomes −4jπl/λ Γ(−l) = ΓLe (12.2.3) For increasing l, one moves away from the load to the generator (or source), l increases, and the phase is decreasing because of the negative sign. So given a point for ΓL on the Smith chart, one has negative phase or decreasing phase by rotating the point clockwise. Also, due to the exp(−4jπl/λ) dependence of the phase, when l = λ/4, the reflection coefficient rotates a half circle around the chart. And when l = λ/2, the reflection coefficient will rotate a full circle, or back to the original point. Therefore, on the edge of the Smith chart, there are indication as to which direction one should rotate if one were to move toward the generator or toward the load. Also, for two points diametrically opposite to each other on the Smith chart, Γ changes sign, and it can be shown easily from (12.2.1) that the normalized impedances are reciprocal of each other. Hence, the Smith chart can also be used to find the reciprocal of a complex number quickly. A full blown Smith chart is shown in Figure 12.5. 128 Electromagnetic Field Theory Figure 12.5: The Smith chart in its full glory. It was invented in 1939 before the age of digital computers, but it still allows engineers to do mental estimations and rough calculations with it, because of its simplicity. 12.3 VSWR (Voltage Standing Wave Ratio) From the previous section, one sees that the voltage and current are not constant in a trans- mission line. Therefore, one surmises that measuring the impedance of a device at microwave frequency is a tricky business. At low frequency, one can use an ohm meter with two wire probes to do such a measurement. But at microwave frequency, two pieces of wire become More on Transmission Lines 129 inductors, and two pieces of metal become capacitors. More sophisticated ways to measure the impedance need to be designed as described below. Due to the interference between that forward traveling wave and the backward traveling wave, V (z) is a function of position z on a terminated transmission line and it is given as −jβz jβz V (z) = V0e + V0e ΓL −jβz 2jβz = V0e 1 + ΓLe −jβz = V0e (1 + Γ(z)) (12.3.1) where we have used (12.1.6) for Γ(z) with γ = jβ. Hence, V (z) is not a constant but dependent on z, or |V (z)| = |V0||1 + Γ(z)| (12.3.2) For lack of a better name, this is called the standing wave, even though it is not truly a standing wave. In Figure 12.6, the relationship between variation of 1 + Γ(z) as z varies is shown. Figure 12.6: The voltage amplitude on a transmission line depends on |V (z)|, which is pro- portional to |1 + Γ(z)| per equation (12.3.2). This figure shows how |1 + Γ(z)| varies as z varies on a transmission line. Using the triangular inequality, one gets the lower and upper bounds or that |V0|(1 − |Γ(z)|) ≤ |V (z)| ≤ |V0|(1 + |Γ(z)|) (12.3.3) But from (12.1.6) and that β is pure real for a lossless line, then |Γ(z)| = |ΓL|; hence Vmin = |V0|(1 − |ΓL|) ≤ |V (z)| ≤ |V0|(1 + |ΓL|) = Vmax (12.3.4) The voltage standing wave ratio, VSWR is defined to be V 1 + |Γ | VSWR = max = L (12.3.5) Vmin 1 − |ΓL| 130 Electromagnetic Field Theory Conversely, one can invert the above to get VSWR − 1 |Γ | = (12.3.6) L VSWR + 1 Hence, the knowledge of voltage standing wave pattern (VSWP), as shown in Figure 12.7, yields the knowledge of |ΓL| the amplitude of ΓL. Notice that the relations between VSWR and |ΓL| are homomorphic to those between Zn and Γ. Therefore, the Smith chart can also be used to evaluate the above equations. Figure 12.7: The voltage standing wave pattern (VSWP) as a function of z on a load- terminated transmission line. The phase of ΓL can also be determined from the measurement of the voltage standing wave pattern. The location of ΓL in Figure 12.6 is determined by the phase of ΓL. Hence, the value of d1 in Figure 12.6 is determined by the phase of ΓL as well. The length of the transmission line waveguide needed to null the original phase of ΓL to bring the voltage standing wave pattern to a maximum value at z = −d1 is shown in Figure 12.7. Thus, d1 is the value where the following equation is satisfied: jφL −4πj(d1/λ) |ΓL|e e = |ΓL| (12.3.7) Therefore, by measuring the voltage standing wave pattern, one deduces both the amplitude and phase of ΓL. From the complex value ΓL, one can determine ZL, the load impedance from the Smith chart. More on Transmission Lines 131 Figure 12.8: A slotted-line equipment which consists of a coaxial waveguide with a slot opening at the top to allow the measurement of the field strength and hence, the voltage standing wave pattern in the waveguide (courtesy of Microwave101.com). In the old days, the voltage standing wave pattern was measured by a slotted-line equip- ment which consists of a coaxial waveguide with a slot opening as shown in Figure 12.8. A field probe can be inserted into the slotted line to determine the strength of the electric field inside the coax waveguide. A typical experimental setup for a slotted line measurement is shown in Figure 12.9. A generator source, with low frequency modulation, feeds microwave energy into the coaxial waveguide. The isolator, allowing only the unidirectional propagation of microwave energy, protects the generator. The attenuator protects the slotted line equip- ment. The wavemeter is an adjustable resonant cavity. When the wavemeter is tuned to the frequency of the microwave, it siphons off some energy from the source, giving rise to a dip in the signal of the SWR meter (a short for voltage-standing-wave-ratio meter). Hence, the wavemeter measures the frequency of the microwave. The slotted line probe is usually connected to a square law detector with a rectifier that converts the microwave signal to a low-frequency signal. In this manner, the amplitude of the voltage in the slotted line can be measured with some low-frequency equipment, such as the SWR meter. Low-frequency equipment is a lot cheaper to make and maintain. That is also the reason why the source is modulated with a low-frequency signal. At low frequencies, circuit theory prevails, engineering and design are a lot simpler. The above describes how the impedance of the device-under-test (DUT) can be measured at microwave frequencies. Nowadays, automated network analyzers make these measurements a lot simpler in a microwave laboratory. More resource on microwave measurements can be found on the web, such as in [90]. Notice that the above is based on the interference of the two traveling wave on a ter- minated transmission line. Such interference experiments are increasingly difficult in optical frequencies because of the much shorter wavelengths. Hence, many experiments are easier to perform at microwave frequencies rather than at optical frequencies. Many technologies are first developed at microwave frequency, and later developed at optical frequency. Examples are phase imaging, optical coherence tomography, and beam steering with phase array sources. Another example is that quantum information and quan- tum computing can be done at optical frequency, but the recent trend is to use artificial atoms 132 Electromagnetic Field Theory working at microwave frequencies. Engineering with longer wavelength and larger component is easier; and hence, microwave engineering. Another new frontier in the electromagnetic spectrum is in the terahertz range. Due to the dearth of sources in the terahertz range, and the added difficulty in having to engineer smaller components, this is an exciting and a largely untapped frontier in electromagnetic technology. Figure 12.9: An experimental setup for a slotted line measurement (courtesy of Pozar and Knapp, U. Mass [91]). Lecture 13 Multi-Junction Transmission Lines, Duality Principle A simple extension of the transmission line theory of the previous lecture is to the case when tranmission lines of different characteristic impedances and wave velocities are concatenated together. Myriads of devices can be designed using such admixture of transmission lines, transistors, and diodes. Also, due to the symmetry of Maxwell’s equations between the electric field and the magnetic field, once a set of solutions have been found for Maxwell’s equations, new solutions can be found by symmetry arguments. This is known as duality principle in electromagnetics. 13.1 Multi-Junction Transmission Lines The real world is usually more complex than the world of our textbooks. However, we need to distill problems in the real world into simpler problems that we can explain with our textbook examples. Figure 13.1 shows many real world technologies, but they can be approximated with transmission line models as shall be seen. 133 134 Electromagnetic Field Theory Figure 13.1: Different kinds of waveguides operating in different frequencies in power lines, RF circuits, microwave circuits, and optical fibers. Their salient physics or features can be captured or approximated by transmission lines (courtesy of Owen Casha). An area where multi-junction transmission lines play an important role is in the microwave integrated circuit (MIC) area and the monolithic microwave integrated circuit (MMIC) area. An MMIC circuit is shown in Figure 13.2. Many of the components can be approximated by multi-junction transmission lines. They are clear motivation for studying this topic. Figure 13.2: A generic GaAs monolithic microwave integrated circuit (MMIC). They are the motivation for studying multi-junction transmission lines (courtesy of Wikipedia). Multi-Junction Transmission Lines, Duality Principle 135 By concatenating sections of transmission lines of different characteristic impedances, a large variety of devices such as resonators, filters, radiators, and matching networks can be formed. We will start with a single junction transmission line first. A good reference for such problem is the book by Collin [92], but much of the treatment here is not found in any textbooks. 13.1.1 Single-Junction Transmission Lines Consider two transmission lines connected at a single junction as shown in Figure 13.3. For simplicity, we assume that the transmission line to the right is infinitely long so that a right- traveling wave is not reflected; namely, there is no reflected wave. And we assume that the two transmission lines have different characteristic impedances, Z01 and Z02. Figure 13.3: A single junction transmission line can be modeled by a equivalent transmission line terminated in a load Zin2. The impedance of the transmission line at junction 1 looking to the right,using the formula from previously derived,1 is −2jβ2l2 1 + ΓL,∞e Zin2 = Z02 −2jβ l = Z02 (13.1.1) 1 − ΓL,∞e 2 2 Since no reflected wave exists, ΓL,∞ = 0, the above is just Z02. As a result, transmission line 1 sees a load of ZL = Zin2 = Z02 hooked to its end. The equivalent circuit is shown in Figure 1We should always remember that the relations between the reflection coefficient Γ and the normalized Zn−1 1+Γ impedance Zn are Γ = and Zn = . Zn+1 1−Γ 136 Electromagnetic Field Theory 13.3 as well. Hence, we deduce that the reflection coefficient at junction 1 between line 1 and line 2, using the knowledge from the previous lecture, is Γ12, and is given by ZL − Z01 Zin2 − Z01 Z02 − Z01 Γ12 = = = (13.1.2) ZL + Z01 Zin2 + Z01 Z02 + Z01 where we have assumed that ZL = Zin2 = Z02. 13.1.2 Two-Junction Transmission Lines Figure 13.4: A single-junction transmission line with a load ZL at the far (load) end of the second line. But it can be reduced to the equivalent circuit shown in the bottom of Figure 13.3. Now, we look at the two-junction case. To this end, we first look at when line 2 is terminated by a load ZL at its load end as shown in Figure 13.4. Then, using the formula derived in the previous lecture, −2jβ2l2 1 + Γ(−l2) 1 + ΓL2e Zin2 = Z02 = Z02 −2jβ l (13.1.3) 1 − Γ(−l2) 1 − ΓL2e 2 2 −2jβ2l2 where we have used the fact that Γ(−l2) = ΓL2e . It is to be noted that here, using knowledge from the previous lecture, that the reflection coefficient at the load end of line 2 is ZL − Z02 ΓL2 = (13.1.4) ZL + Z02 Now, line 1 sees a load of Zin2 hooked at its end. The equivalent circuit is the same as that shown in Figure 13.3. The generalized reflection coefficient at junction 1, which includes all the reflection of waves from its right, is now Zin2 − Z01 Γ˜12 = (13.1.5) Zin2 + Z01 Multi-Junction Transmission Lines, Duality Principle 137 Substituting (13.1.3) into (13.1.5), we have Z ( 1+Γ ) − Z ˜ 02 1−Γ 01 Γ12 = 1+Γ (13.1.6) Z02( 1−Γ ) + Z01 −2jβ2l2 where Γ = ΓL2e . The above can be rearranged to give Z02(1 + Γ) − Z01(1 − Γ) Γ˜12 = (13.1.7) Z02(1 + Γ) + Z01(1 − Γ) Finally, by further rearranging terms, after some somewhat tedious algebra, it can be shown that the above becomes −2jβ2l2 ˜ Γ12 + Γ Γ12 + ΓL2e Γ12 = = −2jβ l (13.1.8) 1 + Γ12Γ 1 + Γ12ΓL2e 2 2 where Γ12, the local reflection coefficient at the junction between line 1 and line 2, is given −2jβ2l2 2 by (13.1.2), and Γ = ΓL2e is the general reflection coefficient at z = −l2 due to the load ZL. In other words, ZL − Z02 ΓL2 = (13.1.9) ZL + Z02 Figure 13.5: A two-junction transmission line with a load ZL at the far end. The input impedance looking in from the far left can be found recursively using the formula (13.1.12) and (13.1.13). Equation (13.1.8) is a powerful formula for multi-junction transmission lines. Imagine now that we add another section of transmission line as shown in Figure 13.5. We can use 2We will use the term “general reflection coefficient” at location z to mean the ratio between the amplitudes of the left-traveling wave and the right-traveling wave on a transmission line. 138 Electromagnetic Field Theory the aforementioned method to first find Γ˜23, the generalized reflection coefficient at junction 2. Using formula (13.1.8), it is given by −2jβ3l3 ˜ Γ23 + ΓL3e Γ23 = −2jβ l (13.1.10) 1 + Γ23ΓL3e 3 3 where ΓL3 is the load reflection coefficient due to the load ZL hooked to the end of transmission line 3 as shown in Figure 13.5. Here, it is given as ZL − Z03 ΓL3 = (13.1.11) ZL + Z03 Given the knowledge of Γ˜23, we can use (13.1.8) again to find the new Γ˜12 at junction 1. It is now −2jβ2l2 Γ12 + Γ˜23e Γ˜12 = (13.1.12) −2jβ l 1 + Γ12Γ˜23e 2 2 The equivalent circuit is again that shown in Figure 13.3. Therefore, we can use (13.1.12) recursively to find the generalized reflection coefficient for a multi-junction transmission line. Once the reflection coefficient is known, the impedance at that location can also be found. For instance, at junction 1, the impedance is now given by 1 + Γ˜12 Zin2 = Z01 (13.1.13) 1 − Γ˜12 instead of (13.1.3). In the above, Z01 is used because the generalized reflection coefficient Γ˜12 is the total reflection coefficient for an incident wave from transmission line 1 that is sent toward the junction 1. Previously, Z02 was used in (13.1.3) because the reflection coefficients in that equation was for an incident wave sent from transmission line 2. If the incident wave were to have come from line 2, then one can write Zin2 as −2jβ2l2 1 + Γ˜23e Zin2 = Z02 (13.1.14) −2jβ l 1 − Γ˜23e 2 2 With some algebraic manipulation, it can be shown that (13.1.13) and (13.1.14) are identical. But (13.1.13) is closer to an experimental scenario where one measures the reflection coefficient by sending a wave from line 1 with no knowledge of what is to the right of junction 1. Transmission lines can be made easily in microwave integrated circuit (MIC) by etching or milling. A picture of a microstrip line waveguide or transmission line is shown in Figure 13.6. Multi-Junction Transmission Lines, Duality Principle 139 Figure 13.6: Schematic of a microstrip line with the signal line above, and a ground plane below (left). A strip line with each strip carrying currents of opposite polarity (right). A ground plane is not needed in the second case. 13.1.3 Stray Capacitance and Inductance Figure 13.7: A general microwave integrated circuit with different kinds of elements. The junction between two transmission lines is not as simple as we have assumed. In the real world, or in MIC, the waveguide junction has discontinuities in line width, or shape. This can give rise to excess charge cumulation or constricted current flow. Excess charge gives rise to excess electric field which corresponds to excess electric stored energy. This can be modeled by stray or parasitic capacitances. Alternatively, there could be excess or constricted current flow that give rise to excess magnetic field. Excess magnetic field gives rise to excess magnetic stored energy. This can be modeled by stray or parasitic inductances. Hence, a junction can be approximated by a circuit model as shown in Figure 13.8 to account for these effects. The Smith chart or the method we have outlined above can still be used to solve for the input impedances of a 140 Electromagnetic Field Theory transmission circuit when these parasitic circuit elements are added. Notice that when the frequency is zero or low, these stray capacitances and inductances are negligible, we retrieve the simple junction model. But since their impedance and admittance are jωLs and jωCs, respectively, they are non-negligible and are instrumental in modeling high frequency circuits. Figure 13.8: A junction between two microstrip lines can be modeled with stray junction capacitance and stray inductances. The capacitance is used to account for excess charges at the junction, while the inductances model the excess current at the junction. They are important as the frequency increases. 13.2 Duality Principle Duality principle exploits the inherent symmetry of Maxwell’s equations. Once a set of E and H fields has been found to solve Maxwell’s equations for a certain geometry, another set for a similar geometry can be found by invoking this principle. Maxwell’s equations in the frequency domain, including the fictitious magnetic sources, are ∇ × E(r, ω) = −jωB(r, ω) − M(r, ω) (13.2.1) ∇ × H(r, ω) = jωD(r, ω) + J(r, ω) (13.2.2) ∇ · B(r, ω) = %m(r, ω) (13.2.3) ∇ · D(r, ω) = %(r, ω) (13.2.4) One way to make Maxwell’s equations invariant is to do the following substitutions. E → H, H → −E, D → B, B → −D (13.2.5) M → −J, J → M, %m → −%, % → %m (13.2.6) The above swaps retain the right-hand rule for plane waves. When material media is included, such that D = ε · E, B = µ · H, for anisotropic media, Maxwell’s equations become ∇ × E = −jωµ · H − M (13.2.7) ∇ × H = jωε · E + J (13.2.8) Multi-Junction Transmission Lines, Duality Principle 141 ∇ · µ · H = %m (13.2.9) ∇ · ε · E = % (13.2.10) In addition to the above swaps, one need further to swap the material parameters, namely, µ → ε, ε → µ (13.2.11) 13.2.1 Unusual Swaps3 There are other swaps where seemingly the right-hand rule is not preserved, e.g., E → H, H → E, M → −J, J → −M, (13.2.12) %m → −%, % → −%m, µ → −ε, ε → −µ (13.2.13) The above swaps will leave Maxwell’s equations invariant, but when applied to a plane wave, the right-hand rule seems violated. The deeper reason is that solutions to Maxwell’s equations are not unique, since there is a time-forward as well as a time-reverse solution. In the frequency domain, this shows up in √ the choice of the sign of the k vector where in a plane wave k = ±ω µε. When one does a swap of µ → −ε and ε → −µ, k is still indeterminate, and one can always choose a root where the right-hand rule is retained. 3This section can be skipped on first reading. 142 Electromagnetic Field Theory 13.3 Fictitious Magnetic Currents Figure 13.9: Sketches of the electric field due to an electric dipole and the magnetic field due to a electric current loop. The E and H fields have the same pattern, and can be described by the same formula. Even though magnetic charges or monopoles do not exist, magnetic dipoles do. For instance, a magnet can be regarded as a magnetic dipole. Also, it is believed that electrons have spins, and these spins make electrons behave like tiny magnetic dipoles in the presence of a magnetic field. Also if we form electric current into a loop, it produces a magnetic field that looks like the electric field of an electric dipole. This resembles a magnetic dipole field. Hence, a magnetic dipole can be made using a small electric current loop (see Figure 13.9). Because of these similarities, it is common to introduce fictitious magnetic charges and magnetic currents into Maxwell’s equations. One can think that these magnetic charges always occur in pair and together. Thus, they do not contradict the absence of magnetic monopole. The electric current loops can be connected in series to make a toroidal antenna as shown in Figure 13.10. The toroidal antenna is used to drive a current in an electric dipole. Notice that the toroidal antenna ressembles the primary winding of a transformer circuit. In essence, Multi-Junction Transmission Lines, Duality Principle 143 the toroidal loops, which mimic a magnetic current loop, produces an electric field that will drive current through the electric dipole. This is dual to the fact that an electric current loop produces a magnetic field. Figure 13.10: A toroidal antenna used to drive an electric current through a conducting cylinder of a dipole. It works similarly to a transformer: one can think of them as the primary and secondary turns (windings) of a transformer (courtesy of Q. S. Liu [93]). 144 Electromagnetic Field Theory Lecture 14 Reflection, Transmission, and Interesting Physical Phenomena We have seen the derivation of a reflection coefficient in a transmission line that relates the amplitude of the reflected wave to that of the incident wave. By doing so, we have used a simplified form of Maxwell’s equations, the telegrapher’s equations which are equations in one dimension. Here, we will solve Maxwell’s equations in its full glory, but in order to do so, we will look at a very simple problem of plane wave reflection and transmission at a single plane interface. This will give rise to the Fresnel reflection and transmission coefficients, and embedded in them are interesting physical phenomena. We will study these interesting phenomena as well. (Much of the contents of this lecture can be found in Kong, and also the ECE 350X lecture notes. They can be found in many textbooks, even though the notations can be slightly different [31, 32, 44, 50,54, 65,79, 83,85,86].) 14.1 Reflection and Transmission—Single Interface Case We will derive the plane-wave reflection coefficients for the single interface case between two dielectric media. These reflection coefficients are also called the Fresnel reflection coefficients because they were first derived by Austin-Jean Fresnel (1788-1827). Note that he lived before the completion of Maxwell’s equations in 1865. But when Fresnel derived the reflection coefficients in 1823, they were based on the elastic theory of light; and hence, the formulas are not exactly the same as what we are going to derive (see Born and Wolf, Principles of Optics, p. 40 [58]). The single plane interface, plane wave reflection and transmission problem, with com- plicated mathematics, is homomorphic to the transmission line problem. The complexity comes because we have to keep track of the 3D polarizations of the electromagnetic fields in this case. We shall learn later that the mathematical homomorphism can be used to exploit 145 146 Electromagnetic Field Theory the simplicity of transmission line theory in seeking the solutions to the multiple dielectric interface problems. 14.1.1 TE Polarization (Perpendicular or E Polarization)1 Figure 14.1: A schematic showing the reflection of the TE polarization wave impinging on a dielectric interface. To set up the above problem, the wave in Region 1 can be written as the superposition or sum of the incident plane wave and reflected plane wave. Here, Ei(r) and Er(r) are the incident and reflected plane waves, respectively. The total field is Ei(r) + Er(r) which are the phasor representations of the fields. We assume plane wave polarized in the y direction where the wave vectors are βi =xβ ˆ ix +zβ ˆ iz, βr =xβ ˆ rx − zβˆ rz, βt =xβ ˆ tx +zβ ˆ tz, respectively for the incident, reflected, and transmitted waves. Then, from Section 7.3, we have −jβi·r −jβixx−jβiz z Ei =yE ˆ 0e =yE ˆ 0e (14.1.1) which represents a uniform incident plane wave, and TE −jβr ·r TE −jβrxx+jβrz z Er =yR ˆ E0e =yR ˆ E0e (14.1.2) which is a uniform reflected wave. In Region 2, we only have transmitted plane wave; hence TE −jβt·r TE −jβtxx−jβtz z Et =yT ˆ E0e =yT ˆ E0e (14.1.3) 1 These polarizations are also variously know as TEz, or the s and p polarizations, a descendent from the notations for acoustic waves where s and p stand for shear and pressure waves respectively. Reflection, Transmission, and Interesting Physical Phenomena 147 In the above, the incident wave is known and hence, E0 is known. From (14.1.2) and (14.1.3), RTE and T TE are unknowns yet to be sought. To find them, we need two boundary conditions to yield two equations.2 These boundary conditions are tangential E field continuous and tangential H field continuous, which aren ˆ × E continuous andn ˆ × H continuous conditions at the interface. Imposingn ˆ × E continuous at z = 0, we get −jβixx TE −jβrxx TE −jβtxx E0e + R E0e = T E0e , ∀x (14.1.4) where ∀ implies “for all”. In order for the above to be valid for all x, it is necessary that 3 βix = βrx = βtx, which is also known as the phase matching condition. From the above, by letting βix = βrx = β1 sin θi = β1 sin θr, we obtain that θr = θi or that the law of reflection that the angle of reflection is equal to the angle of incidence. By letting βix = β1 sin θi = βtx = β2 sin θt, we obtain Snell’s law that β1 sin θi = β2 sin θt. (This law of refraction that was also known in the Islamic world in the 900 AD. [94]). The exponential terms or the phase terms on both sides of (14.1.4) are the same. Now, canceling common terms on both sides of the equation (14.1.4), the above simplifies to 1 + RTE = T TE (14.1.5) To imposen ˆ × H continuous, one needs to find the H field using ∇ × E = −jωµH. By letting ∇ with −jβ, then we have H = −jβ × E/(−jωµ) = β × E/(ωµ). By so doing4 β × E β × yˆ zβˆ − xβˆ i i i −jβi·r ix iz −jβi·r Hi = = E0e = E0e (14.1.6) ωµ1 ωµ1 ωµ1 β × E β × yˆ zβˆ +xβ ˆ r r r TE −jβr ·r rx rz TE −jβr ·r Hr = = R E0e = R E0e (14.1.7) ωµ1 ωµ1 ωµ1 β × E β × yˆ zβˆ − xβˆ t t t TE −jβt·r tx tz TE −jβt·r Ht = = T E0e = T E0e (14.1.8) ωµ2 ωµ2 ωµ2 Imposingn ˆ × H continuous or Hx continuous at z = 0, we have β β β iz −jβixx rz TE −jβrxx tz TE −jβtxx − E0e + R E0e = − T E0e (14.1.9) ωµ1 ωµ1 ωµ2 As mentioned before, the phase-matching condition requires that βix = βrx = βtx. The dispersion relation for plane waves requires that in their respective media, 2 2 2 2 2 2 βix + βiz = βrx + βrz = ω µ1ε1 = β1 , medium 1 (14.1.10) 2 2 2 2 βtx + βtz = ω µ2ε2 = β2 , medium 2 (14.1.11) 2Here, we will treat this problem as a boundary value problem where the unknowns are sought from equations obtained from boundary conditions. 3The phase-matching condition can also be proved by taking the Fourier transform of the equation with respect to x. Among the physics community, this is also known as momentum matching, as the wavenumber of a wave is related to the momentum of the particle. 4We note here that field theory is a lot more complicated than transmission line theory. That is the triumph of transmission line theory as well. 148 Electromagnetic Field Theory Since βix = βrx = βtx = βx (14.1.12) the above implies that βiz = βrz = β1z (14.1.13) Moreover, βtz = β2z 6= β1z usually since β1 6= β2. Then (14.1.9) simplifies to β β 1z (1 − RTE) = 2z T TE (14.1.14) µ1 µ2 p 2 2 p 2 2 where β1z = β1 − βx, and β2z = β2 − βx. Solving (14.1.5) and (14.1.14) for RTE and T TE yields , β β β β RTE = 1z − 2z 1z + 2z (14.1.15) µ1 µ2 µ1 µ2 , β β β T TE = 2 1z 1z + 2z (14.1.16) µ1 µ1 µ2 14.1.2 TM Polarization (Parallel or H Polarization)5 Figure 14.2: A similar schematic showing the reflection of the TM polarization wave impinging on a dielectric interface. The solution to this problem can be easily obtained by invoking duality principle. 5 Also known as TMz polarization. Reflection, Transmission, and Interesting Physical Phenomena 149 The solution to the TM polarization case can be obtained by invoking duality principle where we do the substitution E → H, H → −E, and µ ε as shown in Figure 14.2. The reflection coefficient for the TM magnetic field is then β β β β RTM = 1z − 2z 1z + 2z (14.1.17) ε1 ε2 ε1 ε2 β β β T TM = 2 1z 1z + 2z (14.1.18) ε1 ε1 ε2 Please remember that RTM and T TM are reflection and transmission coefficients for the magnetic fields, whereas RTE and T TE are those for the electric fields. Some textbooks may define these reflection coefficients based on electric field only, and they will look different, and duality principle cannot be applied. 14.2 Interesting Physical Phenomena Three interesting physical phenomena emerge from the solutions of the single-interface prob- lem. They are total internal reflection, Brewster angle effect, and surface plasmonic resonance. We will look at them next. 150 Electromagnetic Field Theory 14.2.1 Total Internal Reflection Figure 14.3: Wave-number surfaces in two regions showing the phase matching condition. The wave number in medium t is larger than the wave number in medium 0. The wave vectors for the incident wave, reflected wave, and transmitted wave have to be aligned in such a way that their components parallel to the interface are equal in order to satisfy the phase-matching condition. One can see that Snell’s law is satisfied when the phase-matching condition is satisfied. Total internal reflection comes about because of phase matching (also called momentum matching). This phase-matching condition can be illustrated using β-surfaces (same as k- surfaces in some literature), as shown in Figure 14.3. It turns out that because of phase matching, for certain interfaces, β2z becomes pure imaginary. Reflection, Transmission, and Interesting Physical Phenomena 151 Figure 14.4: Wave-number surfaces in two regions showing the phase matching condition. The wave number in medium t is smaller than the wave number in medium 0. The figure shows in incident wave vector coming in at the critical angle. Then the transmitted wave vector is parallel to the interface as shown. When the incident angle is larger than the critical angle, βz becomes an imaginary number the wave vector in Region t is complex and cannot be drawn. 2 2 As shown in Figures 14.3 and 14.4, because of the dispersion relation that βrx + βrz = 2 2 2 2 2 2 βix + βiz = β1 , βtx + βtz = β2 , they are equations of two circles in 2D whose radii are β1 and β2, respectively. (The tips of the β vectors for Regions 1 and 2 have to be on a spherical surface in the βx, βy, and βz space in the general 3D case, but in this figure, we only show a cross section of the sphere assuming that βy = 0.) Phase matching implies that the x-component of the β vectors are equal to each other as shown. One sees that θi = θr in Figure 14.4, and also as θi increases, θt increases. For an optically less dense medium where β2 < β1, according to the Snell’s law of refraction, the transmitted β will refract away from the normal, as seen in the figure (where k is synonymous with our β). Therefore, eventually the vector β becomes parallel to the x axis when β = √ t ix βrx = β2 = ω µ2ε2 and θt = π/2. The incident angle at which this happens is termed the critical angle θc (see Figure 14.4). Since βix = β1 sin θi = βrx = β1 sin θr = β2, or √ β2 µ2ε2 n2 sin θr = sin θi = sin θc = = √ = (14.2.1) β1 µ1ε1 n1 √ √ where n1 is the reflective index defined as c0/vi = µiεi/ µ0ε0 where vi is the phase velocity 152 Electromagnetic Field Theory of the wave in Region i. Hence, −1 θc = sin (n2/n1) (14.2.2) p 2 2 When θi > θc. βx > β2 and β2z = β2 − βx becomes pure imaginary. When β2z becomes pure imaginary, the wave cannot propagate in Region 2, or β2z = −jα2z, and the wave becomes evanescent. The reflection coefficient (14.1.15) becomes of the form RTE = (A − jB)/(A + jB) (14.2.3) Since the numerator is the complex conjugate of the denominator. It is clear that |RTE| = 1 and that RTE = ejθTE . Therefore, a total internally reflected wave suffers a phase shift. A phase shift in the frequency domain corresponds to a time delay in the time domain. Such a time delay is achieved by the wave traveling laterally in Region 2 before being refracted back to Region 1. Such a lateral shift is called the Goos-Hanschen shift as shown in Figure 14.5 [58]. A wave that travels laterally along the surface of two media is also known as lateral waves [95,96]. (Please be reminded that total internal reflection comes about entirely due to the phase- matching condition when Region 2 is a faster medium than Region 1. Hence, it will occur with all manner of waves, such as elastic waves, sound waves, seismic waves, quantum waves etc.) Figure 14.5: Goos-Hanschen Shift. A phase delay is equivalent to a time delay (courtesy of Paul R. Berman (2012), Scholarpedia, 7(3):11584 [97]). The guidance of a wave in a dielectric slab is due to total internal reflection at the dielectric- to-air interface. The wave bounces between the two interfaces of the slab, and creates evanes- cent waves outside, as shown in Figure 14.6. The guidance of waves in an optical fiber works by similar mechanism of total internal reflection, as shown in Figure 14.7. Due to the tremen- dous impact the optical fiber has on modern-day communications, Charles Kao, the father of the optical fiber, was awarded the Nobel Prize in 2009. His work was first published in [98]. Reflection, Transmission, and Interesting Physical Phenomena 153 Figure 14.6: The total internal reflections at the two interfaces of a thin-film waveguide can be used to guide an optical wave (courtesy of E.N. Glytsis, NTUA, Greece [99]). Figure 14.7: An optical fiber consists of a core and a cladding region. Total internal reflections occur at the core-cladding interface. They can guide an optical wave in the fiber (courtesy of Wikepedia [100]). Waveguides have affected international communications for over a hundred year now. Since telegraphy was in place before the full advent of Maxwell’s equations, submarine cables for global communications were laid as early as 1850’s. Figure 14.8 shows a submarine cable from 1869 using coaxial cable, and one used in the modern world using optical fiber. 154 Electromagnetic Field Theory Figure 14.8: The picture of an old 1869 submarine cable made of coaxial cables (left), and modern submarine cable made of optical fibers (right) (courtesy of Atlantic-Cable [101], and Wikipedia [102]). Lecture 15 More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines Though simple that it looks, embedded in the TM Fresnel reflection coefficient are a few more interesting physical phenomena. We have looked at the physics of total internal reflection, which has inspired many interesting technologies such as waveguides, the most important of which is the optical fiber. In this lecture, we will look at other physical phenomena. These are the phenomena of Brewster’s angle [103, 104] and that of surface plasmon resonance, or polariton [105, 106]. Even though transmission line theory and the theory of plane wave reflection and trans- mission look quite different, they are very similar in their underlying mathematical structures. For lack of a better name, we call this mathematical homomorphism (math analogy).1 Later, to simplify the mathematics of waves in layered media, we will draw upon this mathemati- cal homomorphism between multi-section transmission line theory and plane-wave theory in layered media. 15.1 Brewster’s Angle We will continue with understanding some interesting phenomena associated with the single- interface problem starting with the Brewster’s angle. 1The use of this term could be to the chagrin of a math person, but it has also being used in a subject called homomorphic encryption or computing [107]. 155 156 Electromagnetic Field Theory Figure 15.1: A figure showing a plane wave being reflected and transmitted at the Brewster’s angle. In Region t, the polarization current or dipoles are all pointing in the βr direction, and hence, there is no radiation in that direction. Brewster angle was discovered in 1815 [103, 104]. Furthermore, most materials at optical frequencies have ε2 6= ε1, but µ2 ≈ µ1. In other words, it is hard to obtain magnetic materials at optical frequencies. Therefore, the TM polarization for light behaves differently from TE polarization. Hence, we shall focus on the reflection and transmission of the TM polarization of light, and we reproduce the previously derived TM reflection coefficient here: β β β β RTM = 1z − 2z 1z + 2z (15.1.1) ε1 ε2 ε1 ε2 The transmission coefficient is easily gotten by the formula T TM = 1 + RTM . Observe that for RTM , it is possible that RTM = 0 if ε2β1z = ε1β2z (15.1.2) p 2 2 Squaring the above, making the note that βiz = βi − βx, one gets 2 2 2 2 2 2 ε2 (β1 − βx ) = ε1 (β2 − βx ) (15.1.3) Solving the above, assuming µ1 = µ2 = µ, gives r √ ε1ε2 βx = ω µ = β1 sin θ1 = β2 sin θ2 (15.1.4) ε1 + ε2 More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines157 The latter two equalities come from phase matching at the interface or Snell’s law. Therefore, r r ε2 ε1 sin θ1 = , sin θ2 = (15.1.5) ε1 + ε2 ε1 + ε2 or squaring the above and adding them, 2 2 sin θ1 + sin θ2 = 1, (15.1.6) 2 2 Then, assuming that θ1 and θ2 are less than π/2, and using the identity that cos θ1 +sin θ1 = 2 2 1, we infer that cos θ1 = sin θ2. Then it follows that sin θ2 = cos θ1 (15.1.7) In other words, θ1 + θ2 = π/2 (15.1.8) The above formula can be used to explain why at Brewster angle, no light is reflected back to Region 1. Figure 15.1 shows that the induced polarization dipoles in Region 2 always have their axes aligned in the direction of reflected wave. A dipole does not radiate along its axis, which can be verified heuristically by field sketch and looking at the Poynting vector. Therefore, these induced dipoles in Region 2 do not radiate in the direction of the reflected ∼ ∼ ∼ wave. Notice that when the contrast is very weak meaning that ε1 = ε2, then θ1 = θ2 = π/4, and (15.1.8) is satisfied. TM Because of the Brewster angle effect for TM polarization when ε2 6= ε1, |R | has to go TM TE through a null when θi = θb. Therefore, |R | ≤ |R | as shown in the plots in Figure 15.2. Then when a randomly (or arbitrarily) polarized light is incident on a surface, the polarization where the electric field is parallel to the surface (TE polarization) is reflected more than the polarization where the magnetic field is parallel to the surface (TM polarization). This phenomenon is used to design sun glasses to reduce road surface glare for drivers. For light reflected off a road surface, they are predominantly horizontally polarized with respect to the surface of the road. When sun glasses are made with vertical polarizers,2 they will filter out and mitigate the reflected rays from the road surface to reduce road glare. This phenomenon can also be used to improve the quality of photography by using a polarizer filter as shown in Figure 15.3. 2Defined as one that will allow vertical polarization to pass through. 158 Electromagnetic Field Theory TM TM Figure 15.2: Because |R | has to through a null when θi = θb, therefore, plots of |R | ≤ TE |R | for all θi as shown above. Figure 15.3: Because that TM and TE lights will be reflected differently, polarizer filter can produce remarkable effects on the quality of the photograph by reducing glare [104]. 15.1.1 Surface Plasmon Polariton Surface plasmon polariton occurs for the same mathematical reason for the Brewster angle effect but the physical mechanism is quite different. Many papers and textbooks will introduce this phenomenon from a different angle. But here, we will see it from the Fresnel reflection coefficient for the TM waves. When the denominator of the reflection coefficient RTM is zero, it can become infinite. This is possible if ε2 < 0, which is possible if medium 2 is a plasma medium. In this case, the criterion for the denominator to be zero is −ε2β1z = ε1β2z (15.1.9) When RTM becomes infinite, it implies that a reflected wave exists when there is no incident TM TM wave. Or when Href = HincR , and R = ∞, Hinc can be zero, and then Href can assume More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines159 any value.3 Hence, there is a plasmonic resonance or guided mode existing at the interface without the presence of an incident wave. It is a self-sustaining wave propagating in the x direction, and hence, is a guided mode propagating in the x direction. Solving (15.1.9) after squaring it, as in the Brewster angle case, yields r √ ε1ε2 βx = ω µ (15.1.10) ε1 + ε2 This is the same equation for the Brewster angle except now that ε2 is negative. Even if ε2 < 0, but ε1 + ε2 < 0 is still possible so that the expression under the square root sign (15.1.10) is positive. Thus, βx can be pure real. The corresponding β1z and β2z in (15.1.9) can be pure imaginary as explained below, and (15.1.9) can still be satisfied. This corresponds to a guided wave propagating in the x direction. When this happens, q 1/2 2 1/2 2 2 √ ε2 √ ε1 β1z = β1 − βx = ω µ ε1 1 − = ω µ (15.1.11) ε1 + ε2 ε1 + ε2 p 2 2 Since ε2 < 0, ε2/(ε1+ε2) > 1, then β1z becomes pure imaginary. Moreover, β2z = β2 − βx 2 and β2 < 0 making β2z becomes even a larger imaginary number. This corresponds to a trapped wave (or a bound state) at the interface. The wave decays exponentially in both directions away from the interface and they are evanescent waves. This mode is shown in Figure 15.4, and is the only case in electromagnetics where a single interface can guide a surface wave, while such phenomenon abounds for elastic waves. When one operates close to the resonance of the mode so that the denominator in (15.1.10) is almost zero, then βx can be very large. The wavelength in the x direction becomes very p 2 2 short in this case, and since βiz = βi − βx, then β1z and β2z become even larger imaginary numbers. Hence, the mode becomes tightly confined or bound to the interface, making the confinement of the mode very tight. This evanescent wave is much more rapidly decaying p 2 2 than that offered by the total internal reflection, which is βz = βt − βx where βx is no larger than β1. It portends use in tightly packed optical components, and has caused some excitement in the optics community. 3In other words, infinity times zero is undefined. This is often encountered in a resonance system like an LC tank circuit. Current flows in the tank circuit despite the absence of an exciting or driving voltage. In an ordinary differential equation or partial differential equation without a driving term (source term), such solutions are known as homogeneous solutions (to clarify the potpouri of math terms, homogeneous solutions here refer to a solution with zero source term). In a matrix equation A · x = b without a right-hand side or that b = 0, it is known as a null-space solution. 160 Electromagnetic Field Theory Figure 15.4: Figure showing a surface plasmonic mode propagating at an air-plasma interface. As in all resonant systems, a resonant mode entails the exchange of energies. In the case of surface plasmonic resonance, the energy is exchanged between the kinetic energy of the electrons and the energy store in the electic field (courtesy of Wikipedia [108]). 15.2 Homomorphism of Uniform Plane Waves and Trans- mission Lines Equations Transmission line theory is very simple due to its one-dimensional nature. But the problem of reflection and transmission of plane waves at a planar interface is actually homormophic to that of the transmission line problem. Therefore, the plane waves through layered medium can be mapped into the multi-section transmission line problem due to mathematical homo- morphism between the two problems. Hence, we can kill two birds with one stone: apply all the transmission line techniques and equations that we have learnt to solve for the solutions of waves through layered medium problems.4 For uniform plane waves, since they are proportional to exp(−jβ · r), we know that with ∇ → −jβ, Maxwell’s equations become β × E = ωµH (15.2.1) β × H = −ωεE (15.2.2) for a general isotropic homogeneous medium. We will specialize these equations for different polarizations. 4This treatment is not found elsewhere, and is peculiar to these lecture notes. More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines161 15.2.1 TE or TEz Waves For this, one assumes a TE wave traveling in the z direction with electric field polarized in the y direction, or E =yE ˆ y, H =xH ˆ x +zH ˆ z. Then we have from (15.2.1) βzEy = −ωµHx (15.2.3) βxEy = ωµHz (15.2.4) From (15.2.2), we have βzHx − βxHz = −ωεEy (15.2.5) The above equations involve three variables, Ey, Hx, and Hz. But there are only two variables in the telegrapher’s equations which are V and I. To this end, we will eliminate one of the variables from the above three equations. Then, expressing Hz in terms of Ey from (15.2.4), we can show from (15.2.5) that β2 β H = −ωεE + β H = −ωεE + x E z x y x z y ωµ y 2 2 2 = −ωε(1 − βx/β )Ey = −ωε cos θEy (15.2.6) where βx = β sin θ has been used. Eqns. (15.2.3) and (15.2.6) can be written to look like the telegrapher’s equations by letting −jβz → d/dz to get d E = jωµH (15.2.7) dz y x d H = jωε cos2 θE (15.2.8) dz x y 2 If we let Ey → V , Hx → −I, µ → L, ε cos θ → C, the above is exactly analogous to the telegrapher’s equations. The equivalent characteristic impedance of these equations above is then r L rµ 1 rµ β ωµ Z0 = = = = (15.2.9) C ε cos θ ε βz βz The above ωµ/βz is the wave impedance for a propagating plane wave with propagation direction or the β inclined with an angle θ respect to the z axis. It is analogous to the characteristic impedance Z0 of a transmission line. When θ = 0, the wave impedance becomes the intrinsic impedance of space. A two region, single-interface reflection problem can then be mathematically mapped to a single-junction connecting two-transmission-lines problem discussed in Section 13.1.1. The equivalent characteristic impedances of these two regions are then ωµ1 ωµ2 Z01 = , Z02 = (15.2.10) β1z β2z 162 Electromagnetic Field Theory We can use the above to find Γ12 as given by Z02 − Z01 (µ2/β2z) − (µ1/β1z) Γ12 = = (15.2.11) Z02 + Z01 (µ2/β2z) + (µ1/β1z) The above is the same as the Fresnel reflection coefficient found earlier for TE waves or RTE after some simple re-arrangement. Assuming that we have a single junction transmission line, one can define a transmission coefficient given by 2Z02 2(µ2/β2z) T12 = 1 + Γ12 = = (15.2.12) Z02 + Z01 (µ2/β2z) + (µ1/β1z) The above is similar to the continuity of the voltage across the junction, which is the same as the continuity of the tangential electric field across the interface. It is also the same as the Fresnel transmission coefficient T TE. 15.2.2 TM or TMz Waves For the TM polarization, by invoking duality principle, the corresponding equations are, from (15.2.7) and (15.2.8), d H = −jωεE (15.2.13) dz y x d E = −jωµ cos2 θH (15.2.14) dz x y Just for consistency of units, since electric field is in V m−1, and magnetic field is in A m−1 we may chose the following map to convert the above into the telegrapher’s equations, viz; 2 Ey → V,Hy → I, µ cos θ → L, ε → C (15.2.15) Then, the equivalent characteristic impedance is now r L rµ rµ β β Z = = cos θ = z = z (15.2.16) 0 C ε ε β ωε The above is also termed the wave impedance of a TM propagating wave making an inclined angle θ with respect to the z axis. Notice again that this wave impedance becomes the intrinsic impedance of space when θ = 0. Now, using the reflection coefficient for a single-junction transmission line, and the appro- priate characteristic impedances for the two lines as given in (15.2.16), we arrive at (β2z/ε2) − (β1z/ε1) Γ12 = (15.2.17) (β2z/ε2) + (β1z/ε1) Notice that (15.2.17) has a sign difference from the definition of RTM derived earlier in the last lecture. The reason is that RTM is for the reflection coefficient of magnetic field while More on Interesting Physical Phenomena, Homomorphism, Plane Waves, and Transmission Lines163 Γ12 above is for the reflection coefficient of the voltage or the electric field. This difference is also seen in the definition for transmission coefficients.5 A voltage transmission coefficient can be defined to be 2(β2z/ε2) T12 = 1 + Γ12 = (15.2.18) (β2z/ε2) + (β1z/ε1) But this will be the transmission coefficient for the voltage, which is not the same as T TM which is the transmission coefficient for the magnetic field or the current. Different textbooks may define different transmission coefficients for this polarization. 5This is often the source of confusion for these reflection and transmission coefficients. 164 Electromagnetic Field Theory Lecture 16 Waves in Layered Media Waves in layered media is an important topic in electromagnetics. Many media can be approximated by planarly layered media. For instance, the propagation of radio wave on the earth surface was of interest and first tackled by Sommerfeld in 1909 [109]. The earth can be approximated by planarly layered media to capture the important physics behind the wave propagation. Many microwave components are made by planarly layered structures such as microstrip and coplanar waveguides. Layered media are also important in optics: they can be used to make optical filters such as Fabry-Perot filters. As technologies and fabrication techniques become better, there is an increasing need to understand the interaction of waves with layered structures or laminated materials. 16.1 Waves in Layered Media Figure 16.1: Waves in layered media. A wave entering the medium from above can be multiply reflected before emerging from the top again. Because of the homomorphism between the transmission line problem and the plane-wave reflection by interfaces, we will exploit the simplicity of the transmission line theory to arrive 165 166 Electromagnetic Field Theory at formulas for plane wave reflection by layered media. This treatment is not found in any other textbooks. 16.1.1 Generalized Reflection Coefficient for Layered Media Figure 16.2: The equivalence of a layered medium problem to a transmission line prob- lem. This equivalence is possible even for oblique incidence. For normal incidence, the wave impedance becomes intrinsic impedances (courtesy of J.A. Kong, Electromagnetic Wave The- ory). Because of the homomorphism between transmission line problems and plane waves in layered medium problems, one can capitalize on using the multi-section transmission line formulas for generalized reflection coefficient, which is −2jβ2l2 Γ12 + Γ˜23e Γ˜12 = (16.1.1) −2jβ l 1 + Γ12Γ˜23e 2 2 In the above, Γ12 is the local reflection at the 1,2 junction, whereas Γ˜ij are the generalized reflection coefficient at the i, j interface. For instance, Γ˜12 includes multiple reflections from behind the 1,2 junction. It can be used to study electromagnetic waves in layered media shown in Figures 16.1 and 16.2. Using the result from the multi-junction transmission line, by analogy we can write down the generalized reflection coefficient for a layered medium with an incident wave at the 1,2 interface, including multiple reflections from behind the interface. It is given by −2jβ2z l2 R12 + R˜23e R˜12 = (16.1.2) −2jβ l 1 + R12R˜23e 2z 2 Waves in Layered Media 167 where R12 is the local Fresnel reflection coefficient and R˜ij is the generalized reflection coef- ficient at the i, j interface. Here, l2 is now the thickness of the region 2. In the above, we assume that the wave is incident from medium (region) 1 which is semi-infinite, the general- ized reflection coefficient R˜12 above is defined at the media 1 and 2 interface. It is assumed that there are multiple reflections coming from the right of the 2,3 interface, so that the 2,3 reflection coefficient is the generalized reflection coefficient R˜23. Figure 16.2 shows the case of a normally incident wave into a layered media. For this case, the wave impedance becomes the intrinsic impedance of homogeneous space. 16.1.2 Ray Series Interpretation of Generalized Reflection Coeffi- cient Figure 16.3: The expression of the generalized reflection coefficient into a ray series. Here, l2 = d2 − d1 is the thickness of the slab (courtesy of [110]). For simplicity, we will assume that R˜23 = R23 in this section. By manipulation, one can con- vert the generalized reflection coefficient R˜12 into a form that has a ray physics interpretation. By adding the term 2 −2jβ2z l2 R12R23e to the numerator of (16.1.2), it can be shown to become −2jβ2z l2 2 ˜ R23e (1 − R12) R12 = R12 + −2jβ l (16.1.3) 1 + R12R23e 2z 2 By using the fact that R12 = −R21 and that Tij = 1 + Rij, the above can be rewritten as −2jβ2z l2 ˜ T12T21R23e R12 = R12 + −2jβ l (16.1.4) 1 + R12R23e 2z 2 Then using the fact that (1 − x)−1 = 1 + x + x2 + ... +, the above can be rewritten as ˜ −2jβ2z l2 2 −4jβ2z l2 R12 = R12 + T12R23T21e + T12R23R21T21e + ··· . (16.1.5) The above allows us to elucidate the physics of each of the terms. The first term in the above is just the result of a single reflection off the first interface. The n-th term above 168 Electromagnetic Field Theory is the consequence of the n-th reflection from the three-layer medium (see Figure 16.3). Hence, the expansion of (16.1.2) into 16.1.5 renders a lucid interpretation for the generalized reflection coefficient. Consequently, the series in 16.1.5 can be thought of as a ray series or a geometrical optics series. It is the consequence of multiple reflections and transmissions in region 2 of the three-layer medium. It is also the consequence of expanding the denominator of the second term in (21). Hence, the denominator of the second term in (21) can be physically interpreted as a consequence of multiple reflections within region 2. 16.1.3 Guided Modes from Generalized Reflection Coefficients We shall discuss finding guided waves in a layered medium next using the generalized reflection coefficient. For a general guided wave along the longitudinal direction parallel to the interfaces (x direction in our notation), the wave will propagate in the manner of e−jβxx For instance, the surface plasmon mode that we found previously can be thought of as a wave propagating in the x direction. The wave number in the x direction, βx for the surface plasmon case can be found in closed form, but it is a rather complicated function of frequency. Hence, it has very interesting phase velocity, which is frequency dependent. From Section 15.1.1, we have derived that for a surface plasmon mode, r √ ε1ε2 βx = ω µ = β1 sin θ1 = β2 sin θ2 (16.1.6) ε1 + ε2 Because ε2 is a plasma medium with complex frequency dependence, βx is in general, a complicated function of ω or frequency. Therefore, this wave has very interesting phase and group velocities, which is typical of waveguide modes. Hence, it is prudent to understand what phase and group velocities are before studying waveguide modes in greater detail. 16.2 Phase Velocity and Group Velocity Now that we know how a medium can be frequency dispersive in a complicated fashion as in the Drude-Lorentz-Sommerfeld (DLS) model, we are ready to investigate the difference between the phase velocity and the group velocity 16.2.1 Phase Velocity The phase velocity is the velocity of the phase of a wave. It is only defined for a mono- chromatic signal (also called time-harmonic, CW (constant wave), or sinusoidal signal) at one given frequency. Given a sinusoidal wave signal, e.g., the voltage signal on a transmission line, using phasor technique, its representation in the time domain can be easily found and take the form V (z, t) = V0 cos(ωt − kz + α) h ω i = V cos k t − z + α (16.2.1) 0 k Waves in Layered Media 169 This sinusoidal signal moves with a velocity ω v = (16.2.2) ph k √ where, for example, k = ω µε, inside a simple coax. Hence, √ vph = 1/ µε (16.2.3) But a dielectric medium can be frequency dispersive, or ε(ω) is not a constant but a function of ω as has been shown with the Drude-Lorentz-Sommerfeld model. Therefore, signals with different ω’s will travel with different phase velocities. More bizarre still, what if the coax is filled with a plasma medium where ω 2 ε = ε 1 − p (16.2.4) 0 ω2 Then, ε < ε0 always meaning that the phase velocity given by (16.2.3) can be larger than the velocity of light in vacuum (assuming µ = µ0). Also, ε = 0 when ω = ωp, implying that k = 0; then in accordance to (16.2.2), vph = ∞. These ludicrous observations can be justified or understood only if we can show that information can only be sent by using a wave packet.1 The same goes for energy which can only be sent by wave packets, but not by CW signal; only in this manner can a finite amount of energy be sent. Therefore, it is prudent for us to study the velocity of a wave packet which is not a mono-chromatic signal. These wave packets can only travel at the group velocity as shall be shown, which is always less than the velocity of light. 1In information theory, according to Shannon, the basic unit of information is a bit, which can only be sent by a digital signal, or a wave packet. 170 Electromagnetic Field Theory 16.2.2 Group Velocity Figure 16.4: A Gaussian wave packet can be thought of as a linear superposition of monochro- matic waves of slightly different frequencies. If one Fourier transforms the above signal, it will be a narrow-band signal centered about certain ω0 (courtesy of Wikimedia [111]). Now, consider a narrow band wave packet as shown in Figure 16.4. It cannot be mono- chromatic, but can be written as a linear superposition of many frequencies. One way to express this is to write this wave packet as an integral in terms of Fourier transform, or a summation over many frequencies, namely2 ¢ ∞ V (z, t) = dωV (z, ω)ejωt (16.2.5) −∞ e To make V (z, t) be related to a traveling wave, we assume that V (z, ω) is the solution to the one-dimensional Helmholtz equation3 e d2 V (z, ω) + k2(ω)V (z, ω) = 0 (16.2.6) dz2 e e To derive this equation, one can easily extend the derivation in Section 7.2 to a dispersive medium where V (z, ω) = Ex(z, ω). Alternatively, one can generalize the derivation in Section 11.2 to the case of dispersive transmission lines. For instance, when the co-axial transmission 2The Fourier transform technique is akin to the phasor technique, but different. For simplicity, we will use V (z, ω) to represent the Fourier transform of V (z, t). e 3In this notes, we will use k and β interchangeably for wavenumber. The transmission line community tends to use β while the optics community uses k. Waves in Layered Media 171 2 2 line is filled with a dispersive material, then k = ω µ0ε(ω). Thus, upon solving the above −jkz equation, one obtains that V (z, ω) = V0(ω)e , and ¢ ∞ j(ωt−kz) V (z, t) = dωV0(ω)e (16.2.7) −∞ In the general case, k is a complicated function of ω as shown in Figure 16.5. Figure 16.5: A typical frequency dependent k(ω) albeit the frequency dependence can be more complicated than shown here. Since this is a wave packet, we assume that V0(ω) is narrow band centered about a frequency ω0, the carrier frequency as shown in Figure 16.6. Therefore, when the integral in (16.2.7) is performed, we need only sum over a narrow range of frequencies in the vicinity of ω0. 172 Electromagnetic Field Theory Figure 16.6: The frequency spectrum of V0(ω) which is the Fourier transform of V0(t). Thus, we can approximate the integrand in the vicinity of ω = ω0, in particular, k(ω) by Taylor series expansion, and let dk(ω ) 1 d2k(ω ) k(ω) =∼ k(ω ) + (ω − ω ) 0 + (ω − ω )2 0 + ··· (16.2.8) 0 0 dω 2 0 dω2 To ensure the real-valuedness of (16.2.5), one ensures that −ω part of the integrand is exactly the complex conjugate of the +ω part, or V (z, −ω) = V ∗(z, ω). Thus the integral can be folded to integrate only over the positive frequencye components.e Another way is to sum over only the +ω part of the integral and take twice the real part of the integral. So, for simplicity, we rewrite (16.2.7) as ¢ ∞ j(ωt−kz) V (z, t) = 2 ¢ ∞ dk ∼ j[ω0t−k(ω0)z] j(ω−ω0)t −j(ω−ω0) dω z V (z, t) = 2 It can be seen that the above integral now involves the integral summation over a small range of ω in the vicinity of ω0. By a change of variable by letting Ω = ω − ω0, it becomes ¢ +∆ dk jΩ(t− dk z) F t − z = dΩV0(Ω + ω0)e dω (16.2.12) dω −∆ When Ω ranges from −∆ to +∆ in the above integral, the value of ω ranges from ω0 − ∆ to ω0 + ∆. It is assumed that outside this range of ω, V0(ω) is sufficiently small so that its value can be ignored. The above itself is a Fourier transform integral that involves only the low frequencies of jΩ t− dk z the Fourier spectrum where e ( dω ) is evaluated over small Ω values. Hence, F is a slowly varying function. Moreover, this function F moves with a velocity dω v = (16.2.13) g dk Here, F (t− z ) in fact is the velocity of the envelope in Figure 16.4. In (16.2.10), the envelope vg function F (t − z ) is multiplied by the rapidly varying function vg ej[ω0t−k(ω0)z] (16.2.14) before one takes the real part of the entire function. Hence, this rapidly varying part represents the rapidly varying carrier frequency shown in Figure 16.4. More importantly, this carrier, the rapidly varying part of the signal, moves with the velocity ω0 vph = (16.2.15) k(ω0) which is the phase velocity. 16.3 Wave Guidance in a Layered Media Now that we have understood phase and group velocity, we are at ease with studying the propagation of a guided wave in a layered medium. We have seen that in the case of a surface plasmonic resonance, the wave is guided by an interface because the Fresnel reflection coefficient becomes infinite. This physically means that a reflected wave exists even if an incident wave is absent or vanishingly small. This condition can be used to find a guided mode in a layered medium, namely, to find the condition under which the generalized reflection coefficient (16.1.2) will become infinite.4 16.3.1 Transverse Resonance Condition Therefore, to have a guided mode exist in a layered medium due to multiple bounces, the generalized reflection coefficient becomes infinite, the denominator of (16.1.2) is zero, or that −2jβ2z l2 1 + R12R˜23e = 0 (16.3.1) 4As mentioned previously in Section 15.1.1, this is equivalent to finding a solution to a problem with no driving term (forcing function), or finding the homogeneous solution to an ordinary differential equation or partial differential equation. It is also equivalent to finding the null space solution of a matrix equation. 174 Electromagnetic Field Theory where t is the thickness of the dielectric slab. Since R12 = −R21, the above can be written as −2jβ2z l2 1 = R21R˜23e (16.3.2) The above has the physical meaning that the wave, after going through two reflections at the two interfaces, 21, and 23 interfaces, which are R21 and R˜23, plus a phase delay given by e−2jβ2z l2 , becomes itself again. This is also known as the transverse resonance condition. When specialized to the case of a dielectric slab with two interfaces and three regions, the above becomes −2jβ2z l2 1 = R21R23e (16.3.3) The above can be generalized to finding the guided mode in a general layered medium. It can also be specialized to finding the guided mode of a dielectric slab. Lecture 17 Dielectric Waveguides As mentioned before, the dielectric waveguide shares many salient features with the optical fiber waveguide, one of the most important waveguides of this century. Before we embark on the study of dielectric waveguides, we will revisit the transverse resonance again. The transverse resonance condition allows one to derive the guidance conditions for a dielectric waveguide easily without having to match the boundary conditions at the interface again: The boundary conditions are already used when deriving the Fresnel reflection coefficients, and hence they are embedded in these reflection coefficients and generalized reflection coefficients. Much of the materials in this lecture can be found in [32,44, 83]. 17.1 Generalized Transverse Resonance Condition The guidance conditions, the transverse resonance condition given previously, can also be derived for the more general case. The generalized transverse resonance condition is a powerful condition that can be used to derive the guidance condition of a mode in a layered medium. To derive this condition, we first have to realize that a guided mode in a waveguide is due to the coherent or constructive interference of the waves. This implies that if a plane wave starts at position 1 (see Figure 17.1)1 and is multiply reflected as shown, it will regain its original phase in the x direction at position 5. Since this mode progresses in the z direction, all these waves (also known as partial waves) are in phase in the z direction by the phase matching condition. Otherwise, the boundary conditions cannot be satisfied. That is, waves at 1 and 5 will gain the same phase in the z direction. But, for them to add coherently or interfere coherently in the x direction, the transverse phase at 5 must be the same as 1. Assuming that the wave starts with amplitude 1 at position 1, it will gain a transverse −jβ0xt phase of e when it reaches position 2. Upon reflection at x = x2, at position 3, the wave −jβ0xt becomes R˜+e where R˜+ is the generalized reflection coefficient at the right interface −2jβ0xt of Region 0. Finally, at position 5, it becomes R˜−R˜+e where R˜− is the generalized 1The waveguide convention is to assume the direction of propagation to be z. Since we are analyzing a guided mode in a layered medium, z axis is as shown in this figure, which is parallel to the interfaces. This is different from before. 175 176 Electromagnetic Field Theory Figure 17.1: The transverse resonance condition for a layered medium. The phase of the wave at position 5 should be equal to the transverse phase at position 1 for constructive interference to occur. reflection coefficient at the left interface of Region 0. For constructive interference to occur or for the mode to exist, we require that −2jβ0xt R˜−R˜+e = 1 (17.1.1) The above is the generalized transverse resonance condition for the guidance condition for a plane wave mode traveling in a layered medium. In (17.1.1), a metallic wall has a reflection coefficient of 1 for a TM wave; hence if R˜+ is 1, Equation (17.1.1) becomes −2jβ0xt 1 − R˜−e = 0. (17.1.2) On the other hand, in (17.1.1), a metallic wall has a reflection coefficient of −1, for TE wave, and Equation (17.1.1) becomes −2jβ0xt 1 + R˜−e = 0. (17.1.3) 17.2 Dielectric Waveguide The most important dielectric waveguide of the modern world is the optical fiber, whose invention was credited to Charles Kao [98]. He was awarded the Nobel prize in 2009 [112]. However, the analysis of the optical fiber requires the use of cylindrical coordinates and special functions such as Bessel functions. In order to capture the essence of dielectric waveguides, one can study the slab dielectric waveguide, which shares many salient features with the optical fiber. This waveguide is also used as thin-film optical waveguides (see Figure 17.2). We start with analyzing the TE modes in this waveguide. Dielectric Waveguides 177 Figure 17.2: An optical thin-film waveguide is made by coating a thin dielectric film or sheet on a metallic surface. The wave is guided by total internal reflection at the top interface, and by metallic reflection at the bottom interface. 17.2.1 TE Case Figure 17.3: Schematic for the analysis of a guided mode in the dielectric waveguide. Total internal reflections occur at the top and bottom interfaces. If the waves add coherently, the wave is guided along the dielectric slab. We shall look at the application of the transverse resonance condition to a TE wave guided in a dielectric waveguide. Again, we assume the direction of propagation of the guided mode to be in the z direciton in accordance with convention. Specializing the above equation to the dielectric waveguide shown in Figure 17.3, we have the guidance condition as −2jβ1xd 1 = R10R12e (17.2.1) where d is the thickness of the dielectric slab. Guidance of a mode is due to total internal reflection, and hence, we expect Region 1 to be optically more dense (in terms of optical refractive indices)2 than region 0 and 2. To simplify the analysis further, we assume Region 2 to be the same as Region 0 so that R12 = R10. The new guidance condition is then 2 −2jβ1xd 1 = R10e (17.2.2) 2Optically more dense means higher optical refractive index, or higher dielectric constant. 178 Electromagnetic Field Theory By phase-matching, βz is the same in all the three regions of Figure 17.3. By expressing all the βix in terms of the variable βz, the above is an implicit equation for βz. Also, we assume that ε1 > ε0 so that total internal reflection occurs at both interfaces as the wave bounces around so that β0x = −jα0x. Therefore, for TE polarization, the local, single-interface, or Fresnel reflection coefficient is µ β − µ β µ β + jµ α 0 1x 1 0x 0 1x 1 0x jθTE R10 = = = e (17.2.3) µ0β1x + µ1β0x µ0β1x − jµ1α0x where θTE is the Goos-Hanschen shift for total internal reflection. It is given by −1 µ1α0x θTE = 2 tan (17.2.4) µ0β1x The guidance condition for constructive interference according to (17.2.1) is such that 2θTE − 2β1xd = 2nπ (17.2.5) From the above, dividing it by four, and taking its tangent, we get θ nπ β d tan TE = tan + 1x (17.2.6) 2 2 2 or using (17.2.4) for the left-hand side, µ α nπ β d 1 0x = tan + 1x (17.2.7) µ0β1x 2 2 The above gives rise to β d µ α = µ β tan 1x , n even (17.2.8) 1 0x 0 1x 2 β d −µ α = µ β cot 1x , n odd (17.2.9) 1 0x 0 1x 2 It can be shown that when n is even, the mode profile is even, whereas when n is odd, the mode profile is odd. The above can also be rewritten as µ β d β d α d 0 1x tan 1x = 0x , even modes (17.2.10) µ1 2 2 2 µ β d β d α d − 0 1x cot 1x = 0x , odd modes (17.2.11) µ1 2 2 2 Again, the above equations can be expressed in the βz variable, but they do not have closed form solutions, save for graphical solutions (or numerical solutions). We shall discuss their graphical solutions.3 3This technique has been put together by the community of scholars in the optical waveguide area. Dielectric Waveguides 179 To solve the above graphically, it is best to plot them in terms of one common variable. It turns out the β1x is the simplest common variable to use for graphical solutions. To this 2 2 2 2 2 2 end, using the fact that −α0x = β0 − βz , and that β1x = β1 − βz , eliminating βz from these two equations, one can show that 1 2 2 2 α0x = [ω (µ11 − µ00) − β1x] (17.2.12) Thus (17.2.10) and (17.2.11) become µ β d β d α d 0 1x tan 1x = 0x µ1 2 2 2 s d2 β d2 = ω2(µ − µ ) − 1x , even modes (17.2.13) 1 1 0 0 4 2 µ β d β d α d − 0 1x cot 1x = 0x µ1 2 2 2 s d2 β d2 = ω2(µ − µ ) − 1x , odd modes (17.2.14) 1 1 0 0 4 2 We can solve the above graphically by plotting µ0 β1xd β1xd y1 = tan even modes (17.2.15) µ1 2 2 µ0 β1xd d y2 = − cot β1x odd modes (17.2.16) µ1 2 2 1 " # 2 d2 β d2 α d y = ω2(µ − µ ) − 1x = 0x (17.2.17) 3 1 1 0 0 4 2 2 180 Electromagnetic Field Theory Figure 17.4: A way to solve (17.2.13) and (17.2.14) is via a graphical method. In this method, both the right-hand side and the left-hand side of the equations are plotted on the same plot. The solutions are at the intersection points of these plots. In the above, y3 is the equation of a circle; the radius of the circle is given by 1 d ω(µ − µ ) 2 . (17.2.18) 1 1 0 0 2 The solutions to (17.2.13) and (17.2.14) are given by the intersections of y3 with y1 and y2. We note from (17.2.1) that the radius of the circle can be increased in three ways: (i) by increasing the frequency, (ii) by increasing the contrast µ11 , and (iii) by increasing the µ00 thickness d of the slab.4 The mode profiles of the first two modes are shown in Figure 17.5. 4These area important salient features of a dielectric waveguide. These features are also shared by the optical fiber. Dielectric Waveguides 181 Figure 17.5: Mode profiles of the TE0 and TE1 modes of a dielectric slab waveguide (courtesy of J.A. Kong [32]). When β0x = −jα0x, the reflection coefficient for total internal reflection is TE µ0β1x + jµ1α0x −1 µ1α0x R10 = = exp +2j tan (17.2.19) µ0β1x − jµ1α0x µ0β1x TE and R10 = 1. Hence, the wave is guided by total internal reflections. Cut-off occurs when the total internal reflection ceases to occur, i.e. when the frequency decreases such that α0x = 0. From Figure 17.4, we see that α0x = 0 when 1 d mπ ω(µ − µ ) 2 = , m = 0, 1, 2, 3,... (17.2.20) 1 1 0 0 2 2 or mπ ωmc = 1 , m = 0, 1, 2, 3,... (17.2.21) d(µ11 − µ00) 2 The mode that corresponds to the m-th cut-off frequency above is labeled the TEm mode. Thus TE0 mode is the mode that has no cut-off or propagates at all frequencies. This is shown in Figure 17.6 where the TE mode profiles are similar since they are dual to each other. The boundary conditions at the dielectric interface is that the field and its normal derivative have to be continuous. The TE0 or TM0 mode can satisfy this boundary condition at all frequencies, but not the TE1 or TM1 mode. At the cut-off frequency, the field outside the slab has to become flat implying the α0x = 0 implying no guidance. 182 Electromagnetic Field Theory Figure 17.6: Mode profiles of the TM modes of a dielectric slab. The TE modes are dual to the TM modes and have similar mode profiles. Next, we will elucidate more physics of the dielectric slab guided mode. At cut-off, α0x = 0, 2 2 2 and from the dispersion relation that α0x = βz − β0 , √ βz = ω µ00, for all the modes. Hence, the phase velocity, ω/βz, and the group velocity, dω/dβz are that of the outer region. This is because when α0x = 0, the wave is not evanescent outside, and the energy of the mode is predominantly carried by the exterior field. When ω → ∞, the radius of the circle in the plot of y3 becomes increasingly larger. As nπ seen from Figure 17.4, the solution for β1x → d for all the modes. From the dispersion relation for Region 1, q 2 2 √ βz = ω µ11 − β1x ≈ ω µ11, ω → ∞ (17.2.22) 2 2 2 since ω µ11 β1x = (nπ/d) . Hence the group and phase velocities approach that of the dielectric slab. This is because when ω → ∞, α0x → ∞, implying the rapid exponential decay of the fields outside the waveguide. Therefore, the fields are trapped or confined in the slab and propagating within it. Because of this, the dispersion diagram of the different modes 5 appear as shown in Figure 17.7. In this figure, kc1, kc2, and kc3 are the cut-off wave number or frequency of the first three modes. Close to cut-off, the field is traveling mostly outside the √ waveguide, and kz ≈ ω µ0ε0. Hence, both the phase and group velocities approach that of 5Please note again that in this course, we will use β and k interchangeably for wavenumbers. Dielectric Waveguides 183 the outer medium as shown in the figure. When the frequency increases, the mode is tightly √ confined in the dielectric slab, and hence, kz ≈ ω µ1ε1. Both the phase and group velocities approach that of Region 1 as shown. Figure 17.7: Here, we have kz versus k1 plots for dielectric slab waveguide. Near its cut-off, the energy of the mode is in the outer region, and hence, its group velocity is close to that of the outer region. At high frequencies, the mode is tightly bound to the slab, and its group velocity approaches that of the dielectric slab (courtesy of J.A. Kong [32]). 17.2.2 TM Case For the TM case, a similar guidance condition analogous to (17.2.1) can be derived but with the understanding that the reflection coefficients in (17.2.1) are now TM reflection coefficients. Similar derivations show that the above guidance conditions, for 2 = 0, µ2 = µ0, reduce to s 2 2 0 d d 2 d d β1x tan β1x = ω (µ11 − µ00) − β1x , even modes (17.2.23) 1 2 2 4 2 s 2 2 0 d d 2 d d − β1x cot β1x = ω (µ11 − µ00) − β1x , odd modes (17.2.24) 1 2 2 4 2 184 Electromagnetic Field Theory Note that for equation (17.2.1), when we have two parallel metallic plates, RTM = 1, and RTE = −1, and the guidance condition becomes mπ 1 = e−2jβ1xd ⇒ β = , m = 0, 1, 2,..., (17.2.25) 1x d These are just the guidance conditions for parallel plate waveguides. 17.2.3 A Note on Cut-Off of Dielectric Waveguides The concept of cut-off in dielectric waveguides is quite different from that of hollow waveguides that we shall learn next. A mode is guided in a dielectric waveguide if the wave is trapped inside, in this case, the dielectric slab. The trapping is due to the total internal reflections at the top and the bottom interfaces of the waveguide. When total internal reflection ceases to occur at any of the two interfaces, the wave is not guided or trapped inside the dielectric slab anymore. This happens when αix = 0 where i can indicate the top-most or the bottom-most region. In other words, the wave ceases to be evanescent in one of the Region i’s. Lecture 18 Hollow Waveguides Hollow waveguides are useful for high-power microwaves. Air has a higher breakdown voltage compared to most materials, and hence, it could be a good medium for propagating high electromagnetic energy. Also, hollow metallic waveguides are sufficiently shielded from the rest of the world so that interference from other sources is minimized. Furthermore, for radio astronomy, they can provide a low-noise system immune to interference. Air generally has less loss than materials, and loss is often the source of thermal noise. A low loss waveguide is also a low noise waveguide.1 18.1 General Information on Hollow Waveguides Many waveguide problems can be solved in closed form. An example is the coaxial waveguide previously discussed. In addition, there are many other waveguide problems that have closed form solutions. Closed form solutions to Laplace and Helmholtz equations are obtained by the separation of variables method. The separation of variables method works only for separable coordinate systems. (There are 11 separable coordinates for Helmholtz equation, but 13 for Laplace equation.) Some examples of separable coordinate systems are cartesian, cylindrical, and spherical coordinates. But these three coordinates are about all we need to know for solving many engineering problems. For other than these three coordinates, complex special functions need to be defined for their solutions, which are hard to compute. Therefore, more complicated cases are now handled with numerical methods using computers. When a waveguide has a center conductor or two conductors like a coaxial cable, it can support a TEM wave where both the electric field and the magnetic field are orthogonal to the direction of propagation. The uniform plane wave is an example of a TEM wave, for instance. However, when the waveguide is hollow or is filled completely with a homogeneous medium, without a center conductor, it cannot support a TEM mode as we shall prove next. 1The fluctuation dissipation theorem [113, 114] says that when a system loses energy to the environment, it also receives the same amount of energy from the environment for energy conservation. In a word, a lossy system loses energy to its environment, but it receives energy back from the environment in terms of thermal noise. 185 186 Electromagnetic Field Theory Much of the materials of this lecture can be found in [32,83, 92]. 18.1.1 Absence of TEM Mode in a Hollow Waveguide Figure 18.1: Absence of TEM mode in a hollow waveguide enclosed by a PEC wall. The magnetic field lines form a closed loop due to the absence of magnetic charges. We would like to prove by contradiction (reductio ad absurdum) that a hollow waveg- uide as shown in Figure 18.1 (i.e. without a center conductor) cannot support a TEM mode as follows. If we assume that TEM mode does exist, then the magnetic field has to end on itself due to the absence of magnetic charges. In this case, it is clear that C Hs ·dl 6= 0 about any closed contour following the magnetic field lines. But Ampere’s law states that the above is equal to ¢ ¢ Hs · dl = jωε E · dS + J · dS (18.1.1) C S S The left-hand side of the above equation is clearly nonzero by the above argument. But for a hollow waveguide, J = 0 and the above becomes ¢ Hs · dl = jωε E · dS (18.1.2) C S Hence, this equation cannot be satisfied unless on the right-hand side there are Ez 6= 0 component. This implies that a TEM mode where both Ez and Hz are zero in a hollow waveguide without a center conductor cannot exist. By the above argument, in a hollow waveguide filled with homogeneous medium, only TEz (TE to z) or TMz (TM to z) modes can exist like the case of a layered medium. For a TEz wave (or TE wave), Ez = 0, Hz 6= 0 while for a TMz wave (or TM wave), Hz = 0, Ez 6= 0. These classes of problems can be decomposed into two scalar problems like the layered medium case, by using the pilot potential method. However, when the hollow waveguide is filled with a center conductor, the TEM mode can exist in addition to TE and TM modes. Hollow Waveguides 187 We begin by studying some simple closed form solutions to hollow waveguides, such as the rectangular waveguides. These closed form solutions offer physical insight into the prop- agation of waves in a hollow waveguide. Another waveguide with slightly more complicated closed form solutions is the circular hollow waveguide. The solutions need to be sought in terms of Bessel functions. Another waveguide with very complicated closed form solutions is the elliptical waveguide. However, the solutions are too complicated to be considered; the preferred method of solving these complicated problems is via numerical method. 18.1.2 TE Case (Ez = 0, Hz 6= 0, TEz case) In this case, the field inside the waveguide is TE to z or TEz. To ensure such a TE field, we can write the E field as E(r) = ∇ × zˆΨh(r) (18.1.3) By construction, equation (18.1.3) will guarantee that Ez = 0. Here, Ψh(r) is a scalar potential andz ˆ is called the pilot vector. 2 The waveguide is assumed source free and filled with a lossless, homogeneous material. Eq. (18.1.3) also satisfies the source-free condition since, clearly, ∇ · E = 0. And hence, from Maxwell’s equations that ∇ × E = jωµH (18.1.4) ∇ × H = −jωεE (18.1.5) it can be shown that ∇ × ∇ × E − ω2µεE = 0 (18.1.6) Furthermore, using the appropriate vector identiy, such as the back-of-the-cab formula, it can be shown that the electric field E(r) satisfies the following Helmholtz wave equation (or partial differential equation) that (∇2 + β2)E(r) = 0 (18.1.7) where β2 = ω2µε. Substituting (18.1.3) into (18.1.7), we get 2 2 (∇ + β )∇ × zˆΨh(r) = 0 (18.1.8) In the above, we assume that ∇2∇ × zˆΨ = ∇ × zˆ(∇2Ψ), or that these operators commute.3 Then it follows that 2 2 ∇ × zˆ(∇ + β )Ψh(r) = 0 (18.1.9) 2It “pilots” the field so that it is transverse to z. 3This is a mathematical parlance, and a commutator is defined to be [A, B] = AB − BA for two operators A and B. If these two operators commute, then [A, B] = 0. 188 Electromagnetic Field Theory Thus, if Ψh(r) satisfies the following Helmholtz wave equation or partial differential equa- tion 2 2 (∇ + β )Ψh(r) = 0 (18.1.10) then (18.1.9) is satisfied, and so is (18.1.7). Hence, the E field constructed with (18.1.3) satisfies Maxwell’s equations, if Ψh(r) satisfies (18.1.10). Figure 18.2: A hollow metallic waveguide with a center conductor (left), and without a center conductor (right). Next, we look at the boundary condition for Ψh(r) which is derivable from the boundary condition for E. The boundary condition for E is thatn ˆ × E = 0 on C, the PEC wall of the waveguide. But from (18.1.3), using the back-of-the-cab (BOTC) formula, nˆ × E =n ˆ × (∇ × zˆΨh) = −nˆ · ∇Ψh = 0 (18.1.11) In applying the BOTC formula, one has to be mindful that ∇ operates on a function to its right, and the function Ψh should be placed to the right of the ∇ operator. ∂ ∂ In the aboven ˆ · ∇ =n ˆ · ∇s where ∇s =x ˆ ∂x +y ˆ∂y (a 2D gradient operator) sincen ˆ has no z component. The boundary condition (18.1.11) then becomes nˆ · ∇sΨh = ∂nΨh = 0 on C (18.1.12) where C is the waveguide wall where ∂n is a shorthand notation forn ˆ · ∇s operator which is a scalar operator. The above is also known as the homogeneous Neumann boundary condition. Furthermore, in a waveguide, just as in a transmission line case, we are looking for traveling wave solutions of the form exp(∓jβzz) for (18.1.10), or that ∓jβz z Ψh(r) = Ψhs(rs)e (18.1.13) Hollow Waveguides 189 where rs =xx ˆ +yy ˆ , or in short, Ψhs(rs) = Ψhs(x, y) is a 2D function. Thus, ∂nΨh = 0 ∂2 2 implies that ∂nΨhs = 0. With this assumption, ∂z2 → −βz , and (18.1.10) becomes even simpler, namely that, 2 2 2 2 2 (∇s + β − βz )Ψhs(rs) = (∇s + βs )Ψhs(rs) = 0 , ∂nΨhs(rs) = 0, on C (18.1.14) 2 2 2 2 2 2 2 2 where ∇s = ∂ /∂x + ∂ /∂y and βs = β − βz . The above is a boundary value problem (BVP) for a 2D waveguide problem. The above 2D wave equation is also called the reduced wave equation. 18.1.3 TM Case (Ez 6= 0, Hz = 0, TMz Case) Repeating similar treatment for TM waves, the TM magnetic field is then H = ∇ × zˆΨe(r) (18.1.15) where 2 2 (∇ + β )Ψe(r) = 0 (18.1.16) We need to derive the boundary condition for Ψe(r) from the fundamental boundary condition thatn ˆ × E = 0 on the waveguide wall. To this end, we find the corresponding E field by taking the curl of the magnetic field in (18.1.15), and thus the E field is proportional to ∂ E ∼ ∇ × ∇ × zˆΨ (r) = ∇∇ · (ˆzΨ ) − ∇2zˆΨ = ∇ Ψ +zβ ˆ 2Ψ (18.1.17) e e e ∂z e e where we have used the BOTC formula to simplify the above. The tangential component of the above isn ˆ × E which is proportional to ∂ nˆ × ∇ Ψ +n ˆ × zβˆ 2Ψ ∂z e e In the above,n ˆ × ∇ is a tangential derivative, and it is clear that the above will be zero if Ψe = 0 on the waveguide wall. Therefore, if Ψe(r) = 0 on C, (18.1.18) where C is the waveguide wall, then, nˆ × E(r) = 0 on C (18.1.19) Equation (18.1.18) is also called the homogeneous Dirichlet boundary condition. Next, we assume that z ∓jβz Ψe(r) = Ψes(rs)e (18.1.20) 2 2 2 This will allow us to replace ∂ /∂z = −βz . Thus, with some manipulation, the boundary value problem (BVP) related to equation (18.1.16) reduces to a simpler 2D problem, i.e., 2 2 (∇s + βs )Ψes(rs) = 0 (18.1.21) with the homogeneous Dirichlet boundary condition that Ψes(rs) = 0, rs on C (18.1.22) To illustrate the above theory, we can solve some simple waveguides problems. 190 Electromagnetic Field Theory 18.2 Rectangular Waveguides Rectangular waveguides are among the simplest waveguides to analyze because closed form solutions exist in cartesian coordinates. One can imagine traveling waves in the xy directions bouncing off the walls of the waveguide causing standing waves to exist inside the waveguide. As shall be shown, it turns out that not all electromagnetic waves can be guided by a hollow waveguide. Only when the wavelength is short enough, or the frequency is high enough that an electromagnetic wave can be guided by a waveguide. When a waveguide mode cannot propagate in a waveguide, that mode is known to be cut-off. The concept of cut-off for hollow waveguide is quite different from that of a dielectric waveguide we have studied previously. 18.2.1 TE Modes (Hz 6= 0, H Modes or TEz Modes) For this mode, the scalar potential Ψhs(rs) satisfies ∂ (∇ 2 + β 2)Ψ (r ) = 0, Ψ (r ) = 0 on C (18.2.1) s s hs s ∂n hs s 2 2 2 4 where βs = β − βz . A viable solution using separation of variables for Ψhs(x, y) is then Ψhs(x, y) = A cos(βxx) cos(βyy) (18.2.2) 2 2 2 where βx + βy = βs . One can see that the above is the representation of standing waves in the xy directions. It is quite clear that Ψhs(x, y) satisfies the BVP and boundary condi- tions defined by equation (18.2.1). Furthermore, cosine functions, rather than sine functions are chosen with the hindsight that the above satisfies the homogenous Neumann boundary condition at x = 0, and y = 0 surfaces. Figure 18.3: The schematic of a rectangular waveguide. By convention, the length of the longer side is usually named a. 4For those who are not familiar with this topic, please consult p. 385 of Kong [32]. Hollow Waveguides 191 To further satisfy the boundary condition at x = a, and y = b surfaces, it is necessary that the boundary condition for eq. (18.1.12) is satisfied or that ∂xΨhs(x, y)|x=a ∼ sin(βxa) cos(βyy) = 0, (18.2.3) ∂yΨhs(x, y)|y=b ∼ cos(βxx) sin(βyb) = 0, (18.2.4) The above puts constraints on βx and βy, implying that βxa = mπ, βyb = nπ where m and n are integers. Hence, (18.2.2) becomes mπ nπ Ψ (x, y) = A cos x cos y (18.2.5) hs a b where mπ 2 nπ 2 β2 + β2 = + = β2 = β2 − β 2 (18.2.6) x y a b s z Clearly, (18.2.5) satisfies the requisite homogeneous Neumann boundary condition at the entire waveguide wall. At this point, it is prudent to stop and ponder on what we have done. Equation (18.2.1) is homomorphic to a matrix eigenvalue problem A · xi = λixi (18.2.7) 2 where xi is the eigenvector and λi is the eigenvalue. Therefore, βs is actually an eigenvalue, and Ψhs(rs) is an eigenfunction (or an eigenmode), which is analogous to an eigenvector. Here, 2 the eigenvalue βs is indexed by m, n, so is the eigenfunction in (18.2.5). The corresponding eigenmode is also called the TEmn mode. 2 The above condition on βs expressed by (18.2.6) is also known as the guidance condition for the modes in the waveguide. Furthermore, from (18.2.6), r mπ 2 nπ 2 β = pβ2 − β2 = β2 − − (18.2.8) z s a b And from (18.2.8), when the frequency is low enough, then mπ 2 nπ 2 β2 = + > β2 = ω2µε (18.2.9) s a b and βz becomes pure imaginary and the mode cannot propagate or becomes evanescent in the z direction.5 For fixed m and n, the frequency at which the above happens is called the cutoff frequency of the TEmn mode of the waveguide. It is given by r 1 mπ 2 nπ 2 ω = √ + (18.2.10) mn,c µε a b 5We have seen this happening in a plasma medium earlier and also in total internal reflection. 192 Electromagnetic Field Theory When ω < ωmn,c, or the wavelength is longer than a certain value, the TEmn mode is evanescent and cannot propagate inside the waveguide. A corresponding cutoff wavelength is then 2 λmn,c = (18.2.11) m 2 n 2 1/2 [ a + b ] So when λ > λmn,c, the mode cannot propagate inside the waveguide. Lowest Guided Mode in a Rectangular Waveguide When m = n = 0, then Ψh(r) = Ψhs(x, y) exp(∓jβzz) is a function independent of x and y. Then E(r) = ∇ × zˆΨh(r) = ∇s × zˆΨh(r) = 0. It turns out the only way for Hz 6= 0 is for H(r) =zH ˆ 0 which is a static field in the waveguide. This is not a very interesting mode, and thus TE00 propagating mode is assumed not to exist and not useful. So the TEmn modes cannot have both m = n = 0. As such, the TE10 mode, when a > b, is the mode with the lowest cutoff frequency or longest cutoff wavelength. Only when the frequency is above this cutoff frequency and the wavelength is shorter than this cutoff wavelength, the TE10 mode can propagate. For the TE10 mode, for the mode to propagate, from (18.2.11), it is needed that λ < λ10,c = 2a (18.2.12) The above has the nice physical meaning that the wavelength has to be smaller than 2a in order for the mode to fit into the waveguide. As a mnemonic, we can think that photons have “sizes”, corresponding to its wavelength. Only when its wavelength is small enough can the photons go into (or be guided by) the waveguide. The TE10 mode, when a > b, is also the mode with the lowest cutoff frequency or longest cutoff wavelength. It is seen with the above analysis, when the wavelength is short enough, or frequency is high enough, many modes can be guided. Each of these modes has a different group and phase velocity. But for most applications, a single guided mode only is desirable. Hence, the knowledge of the cutoff frequencies of the fundamental mode (the mode with the lowest cutoff frequency) and the next higher mode is important. This allows one to pick a frequency window within which only a single mode can propagate in the waveguide. It is to be noted that when a mode is cutoff, the field is evanescent, and there is no real power flow down the waveguide: Only reactive power is carried by such a mode. Lecture 19 More on Hollow Waveguides We have seen that the hollow waveguide is the simplest of waveguides. Closed form solutions exist for such waveguides as seen in the rectangular waveguide case. The solution is elegantly simple and beautiful requiring only trigonometric functions. So we will continue with the study of the rectangular waveguide, and then address another waveguide, the circular waveg- uide where closed form solutions exist also. However, the solution has to be expressed in terms of “Bessel functions”, called special functions. As the name implies, these functions are seldom used outside the context of studying wave phenomena. Bessel functions in cylindrical coordinates are the close cousin of the sinusoidal functions in cartesian coordinates. Whether Bessel functions are more complex or esoteric compared to sinusoidal functions is in the eye of the beholder. Once one is familiarized with them, they are simple. They are also the function that describes the concentric ripple wave that you see in your tea cup every morning (see Figure 19.1)! Figure 19.1: The ripple wave in your tea cup is describable by a Bessel function (courtesy of dreamstime.com). 193 194 Electromagnetic Field Theory 19.1 Rectangular Waveguides, Contd. We have seen the mathematics for the TE modes of a rectangular waveguide. We shall study the TM modes and the modes of a circular waveguide in this lecture. 19.1.1 TM Modes (Ez 6= 0, E Modes or TMz Modes) These modes are not the exact dual of the TE modes because of the boundary conditions. The dual of a PEC (perfect electric conducting) wall is a PMC (perfect magnetic conducting) wall. However, the previous exercise for TE modes can be repeated for the TM modes with caution on the boundary conditions. The scalar wave function (or eigenfunction/eigenmode) for the TM modes, satisfying the homogeneous Dirichlet (instead of Neumann)1 boundary condition with (Ψes(rs) = 0) on the entire waveguide wall is mπ nπ Ψ (x, y) = A sin x sin y (19.1.1) es a b Here, sine functions are chosen for the standing waves, and the chosen values of βx and βy ensure that the boundary condition is satisfied on the x = a and y = b walls. Neither of the m and n can be zero, lest Ψes(x, y) = 0, or the field is zero. Hence, both m > 0, and n > 0 are needed. Thus, the lowest TM mode is the TM11 mode. Thinking of this as an eigenvalue problem, then the eigenvalue is mπ 2 nπ 2 β2 = β2 + β2 = + (19.1.2) s x y a b which is the same as the TE case. Therefore, the corresponding cutoff frequencies and cutoff wavelengths for the TMmn modes are the same as the TEmn modes. Also, these TE and TM modes are degenerate when they share the same eigevalues. Furthermore, the lowest modes, TE11 and TM11 modes have the same cutoff frequency. Figure 19.2 shows the dispersion curves for different modes of a rectangular waveguide. Notice that the group velocities of all the modes are zero at cutoff, and then the group velocities approach that of the waveguide medium as frequency becomes large. These observations can be explained physically. 1Again, “homogeneous” here means “zero”. More on Hollow Waveguides 195 Figure 19.2: Dispersion curves for a rectangular waveguide (courtesy of J.A. Kong [32]). Notice that the lowest TM mode is the TM11 mode, and k is equivalent to β in this course. At cutoff, the guided mode does not propagate in the z direction, and here, the group velocity is zero. But when ω → ∞, the mode propagates in direction almost parallel to the axis of the waveguide, and hence, the group velocity approaches that of the waveguide medium. 19.1.2 Bouncing Wave Picture We have seen that the transverse variation of a mode in a rectangular waveguide can be expanded in terms of sine and cosine functions which represent standing waves which are superposition of two traveling waves, or that they are [exp(−jβxx) ± exp(jβxx)] [exp(−jβyy) ± exp(jβyy)] When the above is expanded and together with the exp(−jβzz) the mode propagating in the z direction in addition to the standing waves in the tranverse direction. Or we see four waves bouncing around in the xy directions and propagating in the z direction. The picture of this bouncing wave can be depicted in Figure 19.3. 196 Electromagnetic Field Theory Figure 19.3: The waves in a rectangular waveguide can be thought of as bouncing waves off the four walls as they propagate in the z direction. 19.1.3 Field Plots Given the knowledge of the scalar piloting potential of a waveguide, one can derive all the field components. For example, for the TE modes, if we know Ψh(r), then E = ∇ × zˆΨh(r), H = −∇ × E/(jωµ) (19.1.3) Then all the electromagnetic field of a waveguide mode can be found, and similarly for TM modes. Plots of the fields of different rectangular waveguide modes are shown in Figure 19.4. Notice that for higher m’s and n’s, with βx = mπ/a and βy = nπ/b, the corresponding βx and βy are larger with higher spatial frequencies. Thus, the transverse spatial wavelengths q 2 2 2 are getting shorter. Also, since βz = β − βx − βy , higher frequencies are needed to make βz real in order to propagate the higher order modes or the high m and n modes. Notice also how the electric field and magnetic field curl around each other. Since ∇×H = jωεE and ∇ × E = −jωµH, they do not curl around each other “immediately” but with a π/2 phase delay due to the jω factor. Therefore, the E and H fields do not curl around each other at one location, but at a displaced location due to the π/2 phase difference. This is shown in Figure 19.5. More on Hollow Waveguides 197 Figure 19.4: Transverse field plots of different modes in a rectangular waveguide (courtesy of Andy Greenwood. Original plots published in Lee, Lee, and Chuang, IEEE T-MTT, 33.3 (1985): pp. 271-274. [115]). Figure 19.5: Field plot of a mode propagating in the z direction of a rectangular waveguide. Notice that the E and H fields do not exactly curl around each other. 198 Electromagnetic Field Theory 19.2 Circular Waveguides Another waveguide where closed-form solutions can be easily obtained is the circular hollow waveguide as shown in Figure 19.6, but they involve the use of Bessel functions. Figure 19.6: Schematic of a circular waveguide in cylindrical coordinates. It is one of the separable coordinate systems. 19.2.1 TE Case 2 For a circular waveguide, it is best first to express the Laplacian operator, ∇s = ∇s · ∇s, in cylindrical coordinates. The first ∇s is a gradient operator while the second ∇s· is a divergence operator: they have different physical meanings. Formulas for grad and div operators are given in many text books [32,116]. Doing a table lookup, ∂ 1 ∂ ∇ Ψ =ρ ˆ Ψ + φˆ s ∂ρ ρ ∂φ 1 ∂ 1 ∂ ∇ · A = ρA + A s ρ ∂ρ ρ ρ ∂φ φ Then 1 ∂ ∂ 1 ∂2 ∇ 2 + β 2 Ψ = ρ + + β 2 Ψ (ρ, φ) = 0 (19.2.1) s s hs ρ ∂ρ ∂ρ ρ2 ∂φ2 s hs The above is the partial differential equation for field in a circular waveguide. It is an eigen- 2 value problem where βs is the eigenvalue, and Ψhs(rs) is the eigenfunction (equivalence of an eigenvector). Using separation of variables, we let ±jnφ Ψhs(ρ, φ) = Bn(βsρ)e (19.2.2) More on Hollow Waveguides 199 ∂2 2 Then ∂φ2 → −n , and (19.2.1) simplifies to an ordinary differential equation which is 1 d d n2 ρ − + β 2 B (β ρ) = 0 (19.2.3) ρ dρ dρ ρ2 s n s 2 Here, dividing the above equation by βs , we can let βsρ in (19.2.2) and (19.2.3) be x. Then the above can be rewritten as 1 d d n2 x − + 1 B (x) = 0 (19.2.4) x dx dx x2 n The above is known as the Bessel equation whose solutions are special functions denoted as 2 Bn(x). (1) (2) These special functions are Jn(x), Nn(x), Hn (x), and Hn (x) which are called Bessel, Neumann, Hankel function of the first kind, and Hankel function of the second kind, respec- tively, where n is the order, and x is the argument.3 Since this is a second order ordinary differential equation, it has only two independent solutions. Therefore, two of the four com- monly encountered solutions of Bessel equation are independent. Thus, they can be expressed in terms of each other. Their relationships are shown below:4 1 Bessel, J (x) = [H (1)(x) + H (2)(x)] (19.2.5) n 2 n n 1 Neumann, N (x) = [H (1)(x) − H (2)(x)] (19.2.6) n 2j n n (1) Hankel–First kind, Hn (x) = Jn(x) + jNn(x) (19.2.7) (2) Hankel–Second kind, Hn (x) = Jn(x) − jNn(x) (19.2.8) It can be shown that r (1) 2 jx−j(n+ 1 ) π H (x) ∼ e 2 2 , x → ∞ (19.2.9) n πx r (2) 2 −jx+j(n+ 1 ) π H (x) ∼ e 2 2 , x → ∞ (19.2.10) n πx They correspond to traveling wave solutions when x = βsρ → ∞. Since Jn(x) and Nn(x) are linear superpositions of these traveling wave solutions, they correspond to standing wave (1) (2) solutions. Moreover, Nn(x), Hn (x), and Hn (x) → ∞ when x → 0. Since the field has to be regular when ρ → 0 at the center of the waveguide shown in Figure 19.6, the only viable solution for the hollow waveguide, to be chosen from (19.2.5) to (19.2.9), is that Bn(βsρ) = AJn(βsρ). Thus for a circular hollow waveguide, the eigenfunction or mode is of the form ±jnφ Ψhs(ρ, φ) = AJn(βsρ)e (19.2.11) 2Studied by Friedrich Wilhelm Bessel, 1784-1846. 3 Some textbooks use Yn(x) for Neumann functions. 4Their relations with each other are similar to those between exp(−jx), sin(x), and cos(x). 200 Electromagnetic Field Theory To ensure that the eigenfunction and the eigenvalue are unique, boundary condition for the partial differential equation is needed. The homogeneous Neumann boundary condition,5. or that ∂nΨhs = 0, on the PEC waveguide wall then translates to d J (β ρ) = 0, ρ = a (19.2.12) dρ n s 0 d 6 Defining Jn (x) = dx Jn(x), the above is the same as 0 Jn (βsa) = 0 (19.2.13) The above are the zeros of the derivative of Bessel function and they are tabulated in many 0 7 textbooks and handbooks. The m-th zero of Jn (x) is denoted to be βnm in many books. Plots of Bessel functions and their derivatives are shown in Figure 19.8, and some zeros of Bessel function and its derivative are also shown in Figure 19.9. With this knowledge, the guidance condition for a waveguide mode is then βs = βnm/a (19.2.14) 2 for the TEnm mode. From the above, βs can be obtained which is the eigenvalue of (19.2.1) and (19.2.3). It is a constant independent of frequency. p 2 2 2 Using the fact that βz = β − βs , then βz will become pure imaginary if β is small 2 2 enough (or the frequency low enough) so that β < βs or β < βs. From this, the corresponding cutoff frequency (the frequency below which βz becomes pure imaginary) of the TEnm mode is 1 β ω = √ nm (19.2.15) nm,c µε a When ω < ωnm,c, the corresponding mode cannot propagate in the waveguide as βz becomes pure imaginary. The corresponding cutoff wavelength is 2π λnm,c = a (19.2.16) βnm By the same token, when λ > λnm,c, the corresponding mode cannot be guided by the waveguide. It is not exactly precise to say this, but this gives us the heuristic notion that if wavelength or “size” of the wave or photon is too big, it cannot fit inside the waveguide. 19.2.2 TM Case The corresponding partial differential equation and boundary value problem for this case is 1 ∂ ∂ 1 ∂2 ρ + + β 2 Ψ (ρ, φ) = 0 (19.2.17) ρ ∂ρ ∂ρ ρ2 ∂φ2 s es 5Note that “homogeneous” here means “zero” in math. 6Note that this is a standard math notation, which has a different meaning in some engineering texts. 7Notably, Abramowitz and Stegun, Handbook of Mathematical Functions [117]. An online version is available at [118]. More on Hollow Waveguides 201 with the homogeneous Dirichlet boundary condition, Ψes(a, φ) = 0, on the waveguide wall. The eigenfunction solution is ±jnφ Ψes(ρ, φ) = AJn(βsρ)e (19.2.18) with the boundary condition that Jn(βsa) = 0. The zeros of Jn(x) are labeled as αnm is many textbooks, as well as in Figure 19.9; and hence, the guidance condition is that for the TMnm mode is that α β = nm (19.2.19) s a 2 where the eigenvalue for (19.2.17) is βs which is a constant independent of frequency. With p 2 2 βz = β − βs , the corresponding cutoff frequency is 1 α ω = √ nm (19.2.20) nm,c µε a or when ω < ωnm,c, the mode cannot be guided. The cutoff wavelength is 2π λnm,c = a (19.2.21) αnm with the notion that when λ > λnm,c, the mode cannot be guided. It turns out that the lowest mode in a circular waveguide is the TE11 mode. It is actually a close cousin of the TE10 mode of a rectangular waveguide. This can be gathered by comparing their field plots: these modes morph into each other as we deform the shape of a rectangular waveguide into a circular waveguide. Figure 19.7: Side-by-side comparison of the field plots of the TE10 mode of a rectangular waveguide versus that of the TE11 mode of a circular waveguide. If one is imaginative enough, one can see that the field plot of one mode morphs into that of the other mode. Electric fields are those that have to end on the waveguide walls. 0 Figure 19.8 shows the plots of Bessel function Jn(x) and its derivative Jn(x). Tables in 0 Figure 19.9 show the roots of Jn(x) and Jn(x) which are important for determining the cutoff frequencies of the TE and TM modes of circular waveguides. 202 Electromagnetic Field Theory 0 Figure 19.8: Plots of the Bessel function, Jn(x), and its derivatives Jn(x). The zeros of 2 these functions are used to find the eigenvalue βs of the problem, and hence, the guidance condition. The left figure is for TM modes, while the right figure is for TE modes. Here, 0 Jn(x) = dJn(x)/dx. 0 Figure 19.9: Table 2.3.1 shows the zeros of Jn(x), which are useful for determining the guidance conditions of the TEmn mode of a circular waveguide. On the other hand, Table 2.3.2 shows the zeros of Jn(x), which are useful for determining the guidance conditions of the TMmn mode of a circular waveguide. More on Hollow Waveguides 203 Figure 19.10: Transverse field plots of different modes in a circular waveguide (courtesy of Andy Greenwood. Original plots published in Lee, Lee, and Chuang [115]). The axially symmetric TE01 mode has the lowest loss, and finds a number of real-world applications as in radio astronomy. 204 Electromagnetic Field Theory Lecture 20 More on Waveguides and Transmission Lines Waveguide is a fundamental component of microwave circuits and systems. The study of closed form solutions offers us physical insight. One can use such insight to design more complex engineering systems. We will use heuristics to understand some systems whose designs follow from physical insight of simpler systems. Also, we will show that the waveguide problem is homomorphic to the transmission line problem. Here again, many transmission line techniques can be used to solve some complex waveguide problems encountered in microwave and optical engineering by adding junction capacitances and inductances. 20.1 Circular Waveguides, Contd. As in the rectangular waveguide case, the guidance of the wave in a circular waveguide can be viewed as bouncing waves in the radial direction. But these bouncing waves give rise to standing waves expressible in terms of Bessel functions. The scalar potential (or pilot potential) for the modes in the waveguide is expressible as ±jnφ Ψαs(ρ, φ) = AJn(βsρ)e (20.1.1) where α = h for TE waves and α = e for TM waves. The Bessel function or wave is expressible in terms of Hankel functions as in (19.2.5). Since Hankel functions are traveling waves, Bessel functions represent standing waves. Therefore, the Bessel waves can be thought of as bouncing traveling waves as in the rectangular waveguide case. In the azimuthal direction, one can express e±jnφ as traveling waves in the φ direction, or they can be expressed as cos(nφ) and sin(nφ) which are standing waves in the φ direction. 205 206 Electromagnetic Field Theory 20.1.1 An Application of Circular Waveguide Figure 20.1: Bouncing wave picture of the Bessel wave inside a circular waveguide for the TE01 mode. When a real-world waveguide is made, the wall of the metal waveguide is not made of perfect electric conductor, but with some metal of finite conductivity. Hence, tangential E field is not zero on the wall implying thatn ˆ · (E × H∗) 6= 0. Thus energy can dissipate into the waveguide wall. It turns out that due to symmetry, the TE01 mode of a circular waveguide has the lowest loss of all the waveguide modes including rectangular waveguide modes. Hence, this waveguide mode is of interest to astronomers who are interested in building low-loss and low-noise systems.1 ˆ The TE01 mode has electric field given by E = φEφ. Furthermore, looking at the magnetic field, the current is mainly circumferential flowing in the φ direction. Moreover, by looking at a bouncing wave picture of the guided waveguide mode, this mode has a small component of tangential magnetic field on a waveguide wall: It becomes increasingly smaller as the frequency increases (see Figure 20.1). The reason is that the wave vector for the waveguide becomes increasingly parallel to the axis of the waveguide with a large βz component compared to the 2 βs component. The wave becomes paraxial in the high-frequency limit. The tangential magnetic field needs to be supported by a surface current on the waveguide wall. This implies that the surface current on the waveguide wall becomes smaller as the frequency increases. The wall loss (or copper loss or eddy current loss) of the waveguide, hence, becomes smaller for higher frequencies. In fact, for high frequencies, the TE01 mode has the smallest copper loss of the waveguide modes: It becomes the mode of choice (see Figure 20.2). Waveguides supporting the TE01 modes are used to connect the antennas of the very large array (VLA) for detecting extra-terrestrial signals in radio astronomy [120] as shown in Figure 20.3. 1Low-loss systems are also low-noise due to energy conservation and the fluctuation dissipation theorem [113, 114, 119]. 2 Recall that for a fixed mode, βs is independent of frequency. More on Waveguides and Transmission Lines 207 Figure 20.2: Losses of different modes in a circular waveguide or radius 1.5 cm . It is seen that at high frequencies, the TE01 mode has the lowest loss (courtesy of [121]). Figure 20.3: Picture of the Very Large Array in New Mexico, USA (courtesy of [120]). 208 Electromagnetic Field Theory Figure 20.4 shows two ways of engineering a circular waveguide so that the TE01 mode is enhanced: (i) by using a mode filter that discourages the guidance of other modes but not the TE01 mode, and (ii), by designing corrugated waveguide wall to discourage the flow of axial current and hence, the propagation of the non-TE01 mode. More details of circular waveguides can be found in [121]. Typical loss of a circular waveguide can be as low as 2 dB/km.3 As shall be shown, an open circular waveguide can be made into an aperture antenna quite easily, because the fields of the aperture are axially symmetric. Such antenna is called a horn antenna. Because of this, the radiation pattern of such an antenna is axially symmetric, which can be used to produce axially symmetric circularly polarized (CP) waves. Ways to enhance the TE01 mode are also desirable [122] as shown in Figure 20.5. Figure 20.4: Ways to enhance the TE01 mode in a circular waveguide: (a) Using mode filter that only allows the mode to go through. (b) Use corrugated waveguide to discourage axial current flow. The field plot of the mode is shown in (c). Such waveguide is used in radio astronomy to design the communication links between antennas in a very large array (VLA [120]), or it is used in a circular horn antenna [122]. 3For optical fiber, this figure of merit (FOM) can be lower than 1 dB/km making the optical fiber a darling for long-distance communication. More on Waveguides and Transmission Lines 209 Figure 20.5: Picture of a circular horn antenna where corrugated wall is used to enhance the TE01 mode (courtesy of [123]). 20.2 Remarks on Quasi-TEM Modes, Hybrid Modes, and Surface Plasmonic Modes We have analyzed some simple structures where closed form solutions are available. These simple elegant solutions offer physical insight into how waves are guided, and how they are cutoff from guidance. As has been shown, for some simple waveguides, the modes can be divided into TEM, TE, and TM modes. However, most waveguides are not simple. We will remark on various complexities that arise in real world applications. 20.2.1 Quasi-TEM Modes Figure 20.6: Some examples of practical coaxial-like waveguides are microstrip line and co- planar waveguide (left), and the optical fiber (right). The environments of these waveguides are inhomogeneous media, and hence, a pure TEM mode cannot propagate on these waveg- uides. 210 Electromagnetic Field Theory Many waveguides cannot support a pure TEM mode even when two conductors are present. For example, two pieces of metal make a transmission line, and in the case of a circular coax, a TEM mode can propagate in the waveguide. But most two-metal transmission lines do not support a pure TEM mode: Instead, they support a quasi-TEM mode. In the optical fiber case, when the index contrast of the fiber is very small, the mode is quasi-TEM as it has to degenerate to the TEM case when the contrast is absent. Absence of TEM Modes in Inhomogeneously-Filled Waveguides In the following, we will give physical arguments as to why a pure TEM mode cannot exists in a microstrip line, a coplanar waveguide, and the optical fiber. When a wave is TEM, it is necessary that the wave propagates with the phase velocity of the medium. But when a uniform waveguide has inhomogeneity in between, as shown in Figure 20.6, this is not possible anymore. We can prove this assertion by reductio ad absurdum. Assume only TE wave in a piecewise homogeneous region, then the E field is 1 E = ∇ × ∇ × (ˆzΨe) (20.2.1) jωεi where εi is the permittivity of the region. By doing some algebra, and assume that the field −jβz z is a waveguide mode such that Ψe has e dependence, then one can show that Ez is given by 1 2 2 Ez = (βi − βz )Ψe (20.2.2) jωεi The above derivation is certainly valid in a piecewise homogeneous region. But each of the piecewise homogeneous media can be made arbitrary small, and hence, it is also valid for inhomogeneous media. If this mode becomes TEM, then Ez = 0 and this is possible only if βz = βi. In other words, the phase velocity of the waveguide mode is the same as a plane TEM wave in the same medium. Now assume that a TEM wave exists in both inhomogeneous regions of the microstrip line or all three dielectric regions of the optical fiber in Figure 20.6. Then the phase velocities in the z direction, determined by ω/βz of each region will be ω/βi of the respective region where βi is the wavenumber of the i-th region. Hence, phase matching is not possible, and the boundary condition cannot be satisfied at the dielectric interfaces. Nevertheless, the lumped element circuit model of the transmission line is still a very good model for such a waveguide. If the line capacitance and line inductances of such lines can be estimated, βz can still be estimated. As shall be shown later, circuit theory is valid when the frequency is low, or the wavelength is large compared to the size of the structures. 20.2.2 Hybrid Modes–Inhomogeneously-Filled Waveguides For most inhomogeneously filled waveguides, the modes (eigenmodes or eigenfunctions) inside are not cleanly classed into TE and TM modes, but with some modes that are the hybrid of TE and TM modes. If the inhomogeneity is piecewise constant, some of the equations we have derived before are still valid: In other words, in the homogeneous part (or constant part) of the More on Waveguides and Transmission Lines 211 waveguide filled with piecewise constant inhomogeneity, the fields can still be decomposed into TE and TM fields. But these fields are coupled to each other by the presence of inhomogeneity, i.e., by the boundary conditions requisite at the interface between the piecewise homogeneous regions. Or both TE and TM waves are coupled together and are present simultaneously, and both Ez 6= 0 and Hz 6= 0. Some examples of inhomogeneously-filled waveguides where hybrid modes exist are shown in Figure 20.7. Sometimes, the hybrid modes are called EH or HE modes, as in an optical fiber. Never- theless, the guidance is via a bouncing wave picture, where the bouncing waves are reflected off the boundaries of the waveguides. In the case of an optical fiber or a dielectric waveguide, the reflection is due to total internal reflection. But in the case of metalic waveguides, the reflection is due to the metal walls. Figure 20.7: Some examples of inhomogeneously filled waveguides where hybrid modes exist: (top-left) A general inhomogeneously filled waveguide, (top-right) slab-loaded rectangular waveguides, and (bottom) an optical fiber with core and cladding. 20.2.3 Guidance of Modes Propagation of a plane wave in free space is by the exchange of electric stored energy and magnetic stored energy. So the same thing happens in a waveguide. For example. in the transmission line, the guidance is by the exchange of electric and magnetic stored energy via the coupling between the line capacitance and the line inductance of the line. In this case, the waveguide size, like the cross-section of a coaxial cable, can be made much smaller than the wavelength. In the case of hollow waveguides, the E and H fields are coupled through their space and time variations. Namely, ∇ × E = −jωµH, ∇ × H = jωεE (20.2.3) 212 Electromagnetic Field Theory Hence, the exchange of the energies stored is via the space that stores these energies, like that of a plane wave. These waveguides work only when these plane waves can “enter” the waveguide. Hence, the size of these waveguides has to be about half a wavelength. The surface plasmonic waveguide is an exception in that the exchange is between the electric field energy stored with the kinetic energy stored in the moving electrons in the plasma instead of magnetic energy stored. This form of energy stored is sometimes referred to as coming from kinetic inductance. Therefore, the dimension of the waveguide can be very small compared to wavelength, and yet the surface plasmonic mode can be guided. 20.3 Homomorphism of Waveguides and Transmission Lines Previously, we have demonstrated mathematical homomorphism between plane waves in lay- ered medium and transmission lines. Such homomorphism can be further extended to waveg- uides and transmission lines. But unlike the plane wave in layered medium case, we cannot replace the ∇ operator with −jβ in a waveguide. Hence, the mathematics is slightly more elaborate. We can show this first for TE modes in a hallow waveguide, and the case for TM modes can be established by invoking duality principle.4 20.3.1 TE Case For this case, Ez = 0, and from Maxwell’s equations ∇ × H = jωεE (20.3.1) ∂ By letting ∇ = ∇s + ∇z, H = Hs + Hz where ∇z =z ˆ∂z , and Hz =zH ˆ z, and the subscript s implies transverse to z components, then (∇s + ∇z) × (Hs + Hz) = ∇s × Hs + ∇z × Hs + ∇s × Hz (20.3.2) where it is understood that ∇z × Hz = 0. Notice that the first term on the right-hand side of the above is pointing in the z direction. Therefore, by letting E = Es + Ez, and equating transverse components in (20.3.1), we have5 ∇z × Hs + ∇s × Hz = jωεEs (20.3.3) To simplify the above equation, we shall remove Hz from above. Next, from Faraday’s law, we have ∇ × E = −jωµH (20.3.4) Again, by letting E = Es + Ez, we can let (20.3.4) be written as ∇s × Es + ∇z × Es + ∇s × Ez = −jωµ(Hs + Hz) (20.3.5) 4I have not seen exposition of such mathematical homomorphism elsewhere except in very simple cases [32]. 5 And from the above, it is obvious that ∇s × Hs = jωεEz, but this equation will not be used in the subsequent derivation. More on Waveguides and Transmission Lines 213 Equating z components of the above, we have ∇s × Es = −jωµHz (20.3.6) The above allows us to express Hz in terms of Es. Using (20.3.6), Eq.(20.3.3) can be rewritten as 1 ∇ × H + ∇ × ∇ × E = +jωεE (20.3.7) z s s −jωµ s s s The above can be further simplified by noting that ∇s × ∇s × Es = ∇s(∇s · Es) − ∇s · ∇sEs (20.3.8) But since ∇ · E = 0, and Ez = 0 for TE modes, it also implies that ∇s · Es = 0. Also, from Maxwell’s equations, we have previously shown that for a homogeneous source-free medium, (∇2 + β2)E = 0 (20.3.9) with Ez = 0 for TE mode, or that 2 2 (∇ + β )Es = 0 (20.3.10) Assuming that we have a guided mode, then ∂2 E ∼ e∓jβz z, E = −β 2E (20.3.11) s ∂z2 s z s Therefore, (20.3.10) becomes 2 2 2 (∇s + β − βz )Es = 0 (20.3.12) or that 2 2 (∇s + βs )Es = 0 (20.3.13) 2 2 2 where βs = β − βz is the transverse wave number. Consequently, from (20.3.8) 2 2 ∇s × ∇s × Es = −∇sEs = βs Es (20.3.14) As such, (20.3.7) becomes 1 ∇ × H = jωεE + ∇ × ∇ × E z s s jωµ s s s 1 = jωεE + β 2E s jωµ s s β 2 β 2 = jωε 1 − s = jωε z E (20.3.15) β2 β2 s 214 Electromagnetic Field Theory Letting βz = β cos θ, then the above can further be rewritten as 2 ∇z × Hs = jωε cos θEs (20.3.16) The above now resembles one of the two telegrapher’s equations that we seek. Now looking at (20.3.4) again, assuming Ez = 0, equating transverse components, we have ∇z × Es = −jωµHs (20.3.17) More explicitly, we can rewrite (20.3.16) and (20.3.17) in the above as ∂ zˆ × H = jωε cos2 θE (20.3.18) ∂z s s ∂ zˆ × E = −jωµH (20.3.19) ∂z s s The above now resembles the telegrapher’s equations. We can multiply (20.3.19) byz ˆ× to get ∂ E = jωµzˆ × H (20.3.20) ∂z s s Now (20.3.18) and (20.3.20) are a set of coupled equations that look even more like the 2 telegrapher’s equations. We can have Es → V ,z ˆ × Hs → −I. µ → L, ε cos θ → C, and the above resembles the telegrapher’s equations, or that the waveguide problem is homomorphic to the transmission line problem. The characteristic impedance of this line is then r L r µ rµ 1 ωµ Z0 = = 2 = = (20.3.21) C ε cos θ ε cos θ βz Therefore, the TE modes of a waveguide can be mapped into a transmission problem. This can be done, for instance, for the TEmn mode of a rectangular waveguide. Then, in the above r mπ 2 nπ 2 β = β2 − − (20.3.22) z a b Therefore, each TEmn mode will be represented by a different characteristic impedance Z0, since βz is different for different TEmn modes. 20.3.2 TM Case This case can be derived using duality principle. Invoking duality, and after some algebra, then the equivalence of (20.3.18) and (20.3.20) become ∂ E = jωµ cos2 θzˆ × H (20.3.23) ∂z s s More on Waveguides and Transmission Lines 215 ∂ zˆ × H = jωεE (20.3.24) ∂z s s 2 To keep the dimensions commensurate, we can let Es → V ,z ˆ × Hs → −I, µ cos θ → L, ε → C, then the above resembles the telegrapher’s equations. We can thus let r r L µ cos2 θ rµ β Z = = = cos θ = z (20.3.25) 0 C ε ε ωε Please note that (20.3.21) and (20.3.25) are very similar to that for the plane wave case, which are the wave impedance for the TE and TM modes, respectively. Figure 20.8: A waveguide filled with layered medium is mathematically homomorphic to a multi-section transmission line problem. Hence, transmission-line method can be used to solve this problem, but junction capacitance and inductance are needed to model the junctions correctly. The above implies that if we have a waveguide of arbitrary cross section filled with layered media, the problem can be mapped to a multi-section transmission line problem, and solved with transmission line methods. When V and I are continuous at a transmission line junction, Es and Hs will also be continuous. Hence, the transmission line solution would also imply continuous E and H field solutions. Figure 20.9: A multi-section waveguide is not exactly homormorphic to a multi-section trans- mission line problem, when the cross section of the waveguides are not equal to each other. Circuit elements are needed at the junctions to capture the physics at the waveguide junctions as shown in the next figure. 216 Electromagnetic Field Theory 20.3.3 Mode Conversion In the waveguide shown in Figure 20.8, there is no mode conversion at the junction interface. Assuming a rectangular waveguide as an example, what this means is that if we send at TE10 into the waveguide, this same mode will propagate throughout the length of the waveguide. The reason is that only this mode alone is sufficient to satisfy the boundary condition at the junction interface. The mode profile does not change throughout the length of the waveguide. To elaborate further, from our prior knowledge, the transverse fields of the waveguide, e.g., for the TM mode, can be derived to be ∓jβz z Hs = ∇ × zˆΨes(rs)e (20.3.26) ∓βz E = ∇ Ψ (r )e∓jβz z (20.3.27) s ωε s es s 2 In the above, βs and Ψes(rs) are eigenvalue and eigenfunction, respectively, that depend only on the geometrical shape of the waveguide, but not the materials filling the waveguide. These eigenfunctions are the same throughout different sections of the waveguide. Therefore, boundary conditions can be easily satisfied at the junctions. However, for a multi-junction waveguide show in Figure 20.9, tangential E and H con- tinuous condition cannot be satisfied by a single mode in each waveguide alone: V and I continuous at a transmission line junction will not guarantee the continuity of tangential E and tangential H fields at the waveguide junction. Multi-modes have to be assumed on both sides of the junction at each section in order to match boundary conditions at the junction [83]. Moreover, mode matching method for multiple modes has to be used at each junction. Typically, a single mode incident at a junction will give rise to multiple modes reflected and multiple modes transmitted. The multiple modes give rise to the phenomenon of mode conversion at a junction. Hence, the waveguide may need to be modeled with multiple transmission lines where each mode is modeled by a different transmission line with different characteristic impedances. However, the operating frequency can be chosen so that only one mode is propagating at each section of the waveguide, and the other modes are cutoff or evanescent. In this case, the multiple modes at a junction give rise to localized energy storage at a junction. These energies can be either inductive or capacitive. The junction effect may be modeled by a simple circuit model as shown in Figure 20.10. These junction elements also account for the physics that the currents and voltages are not continuous anymore across the junction. Moreover, these junction lumped circuit elements account for the stored electric and magnetic energies at the junction. More on Waveguides and Transmission Lines 217 Figure 20.10: Junction circuit elements are used to account for stored electric and magnetic energies at the junction. They also account for that the currents and voltages are not con- tinuous across the junctions anymore as the fields of the dominant modes in each section as shown in Figure 20.9 are not continuous anymore. 218 Electromagnetic Field Theory Lecture 21 Cavity Resonators Cavity resonators are important components of microwave and optical systems. They work by constructive and destructive interference of waves in an enclosed region. They can be used as filters, or as devices to enhance certain physical interactions. These can be radiation antennas or electromagnetic sources such as magnetrons or lasers. They can also be used to enhance the sensitivity of sensors. We will study a number of them, and some of them, only heuristically in this lecture. 21.1 Transmission Line Model of a Resonator The simplest cavity resonator is formed by using a transmission line. The source end can be terminated by ZS and the load end can be terminated by ZL. When ZS and ZL are non- dissipative, such as when they are reactive loads (capacitive or inductive), then no energy is dissipitated as a wave is reflected off them. Therefore, if the wave can bounce and interfere constructively between the two ends, a coherent solution or a resonant solution can exist due to constructive inference. The resonant solution exists even when the source is turned off. In mathematical par- lance, this is a homogeneous solution to a partial differential equation or ordinary differential equation, since the right-hand side of the pertinent equation is zero. The right-hand side of these equations usually corresponds to a source term or a driving term. In physics parlance, this is a natural solution since it exists naturally without the need for a driving or exciting source. 219 220 Electromagnetic Field Theory Figure 21.1: A simple resonator can be made by terminating a transmission line with two reactive loads at its two ends, the source end with ZS and the load end with ZL. The transverse resonance condition for 1D problem can be used to derive the resonance condition, namely that −2jβz d 1 = ΓSΓLe (21.1.1) where ΓS and ΓL are the reflection coefficients at the source and the load ends, respectively, βz the the wave number of the wave traveling in the z direction, and d is the length of the transmission line. For a TEM mode in the transmission line, as in a coax filled with homogeneous medium, then βz = β, where β is the wavenumber for the homogeneous medium. Otherwise, for a quasi-TEM mode, βz = βe where βe is some effective wavenumber for a z- propagating wave in a mixed medium. In general, βe = ω/ve (21.1.2) where ve is the effective phase velocity of the wave in the heterogeneous structure. When the source and load impedances are replaced by short or open circuits, then the reflection coefficients are −1 for a short, and +1 for an open circuit. The (21.1.1) above then becomes ±1 = e−2jβed (21.1.3) The ± sign corresponds to different combinations of open and short circuits at the two ends of the transmission lines. When a “+” sign is chosen, which corresponds to either both ends are short circuit, or are open circuit, the resonance condition is such that βed = pπ, p = 0, 1, 2,..., or integer (21.1.4) For a TEM or a quasi-TEM mode in a transmission line, p = 0 is not allowed as the voltage will be uniformly zero on the transmisson line or V (z) = 0 for all z implying a trivial solution. The lowest mode then is when p = 1 corresponding to a half wavelength on the transmission line. When the line is open at one end, and shorted at the other end in (21.1.1), the resonance condition corresponds to the “−” sign in (21.1.3), which gives rise to βed = pπ/2, p odd (21.1.5) Cavity Resonators 221 The lowest mode is when p = 1 corresponding to a quarter wavelength on the transmission line, which is smaller than that of a transmission line terminated with short or open at both ends. Designing a small resonator is a prerogative in modern day electronic design. For example, miniaturization in cell phones calls for smaller components that can be packed into smaller spaces. A quarter wavelength resonator made with a coax is shown in Figure 21.2. It is easier to make a short indicated at the left end, but it is hard to make a true open circuit as shown at the right end. A true open circuit means that the current has to be zero. But when a coax is terminated with an open, the electric current does end abruptly. The fringing field at the right end gives rise to stray capacitance through which displacement current can flow in accordance to the generalized Ampere’s law. Hence, we have to model the right end termination with a small stray or fringing field capacitance as shown in Figure 21.2. This indicates that the current does not abruptly go to zero at the right-hand side due to the presence of fringing field and hence, displacement current. Figure 21.2: A short and open circuited transmission line can be a resonator, but the open end has to be modeled with a fringing field capacitance Cf since there is no exact open circuit. 21.2 Cylindrical Waveguide Resonators Since a cylindrical waveguide1 is homomorphic to a transmission line, we can model a mode in this waveguide as a transmission line. Then the termination of the waveguide with either a short or an open circuit at its end makes it into a resonator. Again, there is no true open circuit in an open ended waveguide, as there will be fringing fields at its open ends. If the aperture is large enough, the open end of the waveguide radiates and may be used as an antenna as shown in Figure 21.3. 1Both rectangular and circular waveguides are cylindrical waveguides. 222 Electromagnetic Field Theory Figure 21.3: A rectangular waveguide terminated with a short at one end, and an open circuit at the other end. The open end can also act as an antenna as it also radiates (courtesy of RFcurrent.com). As previously shown, single-section waveguide resonators can be modeled with a transmis- p 2 2 sion line using homomorphism with the appropriately chosen βz. Then, βz = β − βs where βs can be found by first solving a 2D waveguide problem corresponding to the reduced-wave equation. For a rectangular waveguide, for example, from previous lecture, r mπ 2 nπ 2 β = β2 − − (21.2.1) z a b 2 for both TEmn and TMmn modes. If the waveguide is terminated with two shorts (which is easy to make) at its ends, then the resonance condition is that βz = pπ/d, p integer (21.2.2) Together, using (21.2.1), we have the condition that ω2 mπ 2 nπ 2 pπ 2 β2 = = + + (21.2.3) c2 a b d The above can only be satisfied by certain select frequencies, and these frequencies are the resonant frequencies of the rectangular cavity. The corresponding mode is called the TEmnp mode or the TMmnp mode depending on if these modes are TE to z or TM to z. 2 It is noted that for a certain mn mode, with a choice of frequency, βz = 0 which does not happen in a transmission line. Cavity Resonators 223 The entire electromagnetic fields of the cavity can be found from the scalar potentials previously defined, namely that E = ∇ × zˆΨh, H = ∇ × E/(−jω) (21.2.4) H = ∇ × zˆΨe, E = ∇ × H/(jωε) (21.2.5) Figure 21.4: A waveguide filled with layered dielectrics can also become a resonator. The transverse resonance condition can be used to find the resonant modes. Since the layered medium problem in a waveguide is the same as the layered medium problem in open space, we can use the generalized transverse resonance condition to find the resonant modes of a waveguide cavity loaded with layered medium as shown in Figure 21.4. This condition is repeated below as: −2jβz d R˜−R˜+e = 1 (21.2.6) where d is the length of the waveguide section where the above is applied, and R˜− and R˜+ are the generalized reflection coefficient to the left and right of the center waveguide section. The above is similar to the resonant condition using the transmission line model in (21.1.1), except that now, we have replaced the transmission line reflection coefficient with TE or TM generalized reflection coefficients. 21.2.1 βz = 0 Case In this case, we can still look at the TE and the TM modes in the waveguide. This corresponds to a waveguide mode that bounces off the waveguide wall, but make no progress in the z direction. The modes are independent of z since βz = 0. It is quite easy to show that for the TE case, a z-independent H =zH ˆ 0, and E = Es exist inside the waveguide, and for the TM case, a z-independent E =zE ˆ 0, and H = Hs being the only components in the waveguide. Consider now a single section waveguide. For the TE mode, if either one of the ends of the waveguide is terminated with a PEC wall, thenn ˆ · H = 0 at the end. This will force the z-independent H field to be zero in the entire waveguide. Thus for the TE mode, it can only exist if both ends are terminated with open, but this mode is not trapped inside since it easily leaks energy to the outside via the ends of the waveguide. For the TM mode, since E =zE ˆ 0, it easily satisfy the boundary condition if both ends are terminated with PEC walls since the boundary condition is thatn ˆ × E = 0. The wonderful part about this mode is that the length or d of the cavity can be as short as possible. 224 Electromagnetic Field Theory 21.2.2 Lowest Mode of a Rectangular Cavity The lowest TM mode in a rectanglar waveguide is the TM11 mode. At the cutoff of this mode, the βz = 0 or p = 0, implying no variation of the field in the z direction. When the two ends are terminated with metallic shorts, the tangential magnetic field is not shorted out. But the tangential electric field is shorted to zero in the entire cavity, or that the TE mode cannot exist. Therefore, the longitudinal electric field of the TM mode still exists (see Figures 21.5 and 21.6). As such, for the TM mode, m = 1, n = 1 and p = 0 is possible giving a non-zero field in the cavity. This is the TM110 mode of the resonant cavity, which is the lowest mode in the cavity if a > b > d. The top and side views of the fields of this mode is shown in Figures 21.5 and 21.6. The corresponding resonant frequency of this mode satisfies the equation ω2 π 2 π 2 110 = + (21.2.7) c2 a b Figure 21.5: The top view of the E and H fields of a rectangular resonant cavity. Figure 21.6: The side view of the E and H fields of a rectangular resonant cavity (courtesy of J.A. Kong [32]). For the TE modes, it is required that p 6= 0, otherwise, the field is zero in the cavity. For example, it is possible to have the TE101 mode with nonzero E field. The resonant frequency Cavity Resonators 225 of this mode is ω2 π 2 π 2 101 = + (21.2.8) c2 a d Clearly, this mode has a higher resonant frequency compared to the TM110 mode if d < b. The above analysis can be applied to circular and other cylindrical waveguides with βs determined differently. For instance, for a circular waveguide, βs is determined differently using Bessel functions, and for a general arbitrarily shaped waveguide, βs may have to be determined numerically. Figure 21.7: A circular resonant cavity made by terminating a circular waveguide (courtesy of Kong [32]). For a spherical cavity, one would have to analyze the problem in spherical coordinates. The equations will have to be solved by the separation of variables using spherical harmonics. Details are given on p. 468 of Kong [32]. 21.3 Some Applications of Resonators Resonators in microwaves and optics can be used for designing filters, energy trapping devices, and antennas. As filters, they are used like LC resonators in circuit theory. A concatenation of them can be used to narrow or broaden the bandwidth of a filter. As an energy trapping device, a resonator can build up a strong field inside the cavity if it is excited with energy close to its resonance frequency. They can be used in klystrons and magnetrons as microwave sources, a laser cavity for optical sources, or as a wavemeter to measure the frequency of 226 Electromagnetic Field Theory the electromagnetic field at microwave frequencies. An antenna is a radiator that we will discuss more fully later. The use of a resonator can help in resonance tunneling to enhance the radiation efficiency of an antenna. 21.3.1 Filters Microstrip line resonators are often used to make filters. Transmission lines are often used to model microstrip lines in a microwave integrated circuits (MIC)or monolithic MIC (MMIC). In these circuits, due to the etching process, it is a lot easier to make an open circuit rather than a short circuit. But a true open circuit is hard to make as an open ended microstrip line has fringing field at its end as shown in Figure 21.8 [124,125]. The fringing field gives rise to fringing field capacitance as shown in Figure 21.2. Then the appropriate ΓS and ΓL can be used to model the effect of fringing field capacitance. Figure 21.9 shows a concatenation of several microstrip resonators to make a microstrip filter. This is like using a concatenation of LC tank circuits to design filters in circuit theory. Optical filters can be made with optical etalon as in a Fabry-Perot resonator, or concate- nation of them. This is shown in Figure 21.10. Figure 21.8: End effects and junction effects in a microwave integrated circuit [124, 125] (courtesy of Microwave Journal). Cavity Resonators 227 Figure 21.9: A microstrip filter designed using concatenated resonators. The connectors to the coax cable are the SMA (sub-miniature type A) connectors (courtesy of aginas.fe.up.pt). Figure 21.10: Design of a Fabry-Perot resonator [56,83, 126,127]. 21.3.2 Electromagnetic Sources Microwave sources are often made by transferring kinetic energy from an electron beam to microwave energy. Klystrons, magnetrons, and traveling wave tubes are such devices. 228 Electromagnetic Field Theory However, the cavity resonator in a klystron enhances the interaction of the electrons with the microwave field allowing for such energy transfer, causing the field to grow in amplitude as shown in Figure 21.11. Magnetron cavity works also by transferring the kinetic energy of the electron into the microwave energy. By injecting hot electrons into the magnetron cavity, the electromagnetic cavity resonance is magnified by the absorbing kinetic energy from the hot electrons, giving rise to microwave energy. Figure 21.13 shows laser cavity resonator to enhance of light wave interaction with material media. By using stimulated emission of electronic transition, light energy can be produced. Figure 21.11: A klystron works by converting the kinetic energy of an electron beam into the energy of a traveling microwave next to the beam (courtesy of Wiki [128]). Cavity Resonators 229 Figure 21.12: A magnetron works by having a high-Q microwave cavity resonator. When the cavity is injected with energetic electrons from the cathode to the anode, the kinetic energy of the electron feeds into the energy of the microwave (courtesy of Wiki [129]). Figure 21.13: A simple view of the physical principle behind the working of the laser (courtesy of www.optique-ingenieur.org). Energy trapping of a waveguide or a resonator can be used to enhance the efficiency of a semiconductor laser as shown in Figure 21.14. The trapping of the light energy by the heterojunctions as well as the index profile allows the light to interact more strongly with 230 Electromagnetic Field Theory the lasing medium or the active medium of the laser. This enables a semiconductor laser to work at room temperature. In 2000, Z. I. Alferov and H. Kroemer, together with J.S. Kilby, were awarded the Nobel Prize for information and communication technology. Alferov and Kroemer for the invention of room-temperature semiconductor laser, and Kilby for the invention of electronic integrated circuit (IC) or the chip. Figure 21.14: A semiconductor laser at work. Room temperature lasing is possible due to both the tight confinement of light and carriers (courtesy of Photonics.com). 21.3.3 Frequency Sensor A cavity resonator can be used as a frequency sensor. It acts as an energy trap, because it will siphon off energy from a microwave when the microwave frequency hits the resonance frequency of the cavity resonator. This can be used to determine the frequency of the passing wave. Wavemeters are shown in Figures 21.15 and 21.16. As seen in the picture, there is an entry microwave port for injecting microwave into the cavity, and another exit port for the microwave to leave the cavity sensor. The passing microwave, when it hits the resonance frequency of the cavity, will create a large field inside it. The larger field will dissipate more energy on the cavity metallic wall, and gives rise to less energy leaving the cavity. This dip in energy transmission reveals the frequency of the microwave. Cavity Resonators 231 Figure 21.15: An absorption wave meter can be used to measure the frequency of microwave (courtesy of Wiki [130]). Figure 21.16: The innards of a wavemeter (courtesy of eeeguide.com). 232 Electromagnetic Field Theory Lecture 22 Quality Factor of Cavities, Mode Orthogonality Cavity resonators are important for making narrow band filters. The bandwidth of a filter is related to the Q or the quality factor to the cavity. A concatenation of cavity resonators can be used to engineer different filter designs. Resonators can also be used to design various sensing systems, as well as measurement systems. We will study this concept of Q in this lecture. Also, before we leave the lectures on waveguides and resonators, it will be prudent to discuss mode orthogonality. Since this concept is very similar to eigenvector orthogonality found in matrix or linear algebra, we will relate mode orthogonality in waveguides and cavities to eigenvector orthogonality. 22.1 The Quality Factor of a Cavity–General Concept The quality factor of a cavity or its Q measures how ideal or lossless a cavity resonator is. An ideal lossless cavity resonator will sustain free oscillations forever, while most resonators sustain free oscillations for a finite time. This is because of losses coming from radiation, dissipation in the dielectric material filling the cavity, or resistive loss of the metallic part of the cavity. 233 234 Electromagnetic Field Theory 22.1.1 Analogue with an LC Tank Circuit Figure 22.1: For the circuit on the left, it will resonate forever even if the source is turned off. But for the circuit on the right, the current in the circuit will decay with time due to dissipation in the resistor. One of the simplest resonators imaginable is the LC tank circuit. By using it as an analogue, we can better understand the resonance of a cavity. When there is no loss in an LC tank circuit, it can oscillate forever. Moreover, if we turn off the source, a free oscillation solution exists.1 One can write the voltage-current relation in the circuit as V I = = VY (22.1.1) jωL + 1/(jωC) where 1 Y (ω) = (22.1.2) jωL + 1/(jωC) The above Y (ω) can be thought of as the transfer function of the linear system where the in put is V (ω) and the output is I(ω). When the voltage is zero or turned off, a non-zero current exist or persist at the resonance frequency of the oscillator. The resonant frequency is when the denominator in the above equation is zero, so that I is finite despite V = 0. This√ resonant frequency, obtained by setting the denominator of Y to zero, is given by ωR = 1/ LC. When a small resistor is added in the circuit to give rise to loss, the voltage-current relation becomes V I = = VY (22.1.3) jωL + R + 1/(jωC) 1 Y = (22.1.4) jωL + R + 1/(jωC) 1This is analogous to the homogeneous solution of an ordinary differential equation. Quality Factor of Cavities, Mode Orthogonality 235 Now, the denominator of the above functions can never go to zero for real ω. But there exists complex ω that will make Y become infinite. These are the complex resonant frequencies of the circuit. The homogeneous solution (also called the natural solution, or free oscillation) can only exists at the complex resonant frequencies. With complex resonances, the voltage and the current are decaying sinusoids. By the same token, because of losses, the free oscillation in a cavity has electromagnetic field with time dependence as follows: −αt −αt E ∝ e cos(ωt + φ1), H ∝ e cos(ωt + φ2) (22.1.5) That is, they are decaying sinusoids. The total time-average stored energy, which is propor- 1 2 1 2 tional to 4 |E| + 4 µ |H| is of the form . −2αt hWT i = hWEi + hWH i = W0e (22.1.6) If there is no loss, hWT i will remain constant. However, with loss, the average stored energy 1 will decrease to 1/e of its original value at t = τ = 2α . The Q of a cavity is defined as the number of free oscillations in radians (rather than cycles) that the field undergoes before the energy stored decreases to 1/e of its original value (see Figure 22.2). In a time interval 1 ω τ = 2α , the number of free oscillations in radians is ωτ or 2α ; hence, the Q is defined to be [32] . ω Q = (22.1.7) 2α Q is an approximate concept, and makes sense only if the system has low loss. Figure 22.2: A typical time domain response of a high Q system (courtesy of Wikipedia). 236 Electromagnetic Field Theory Furthermore, by energy conservation, the decrease in stored energy per unit time must be equal to the total power dissipated in the losses of a cavity, in other words, dhW i hP i = − T (22.1.8) D dt By further assuming that WT has to be of the form in (22.1.6), then dhWT i . − = 2αW e−2αt = 2αhW i (22.1.9) dt 0 t From the above, we can estimate the decay constant . hPDi α = (22.1.10) 2hWT i Hence, we can write equation (22.1.7) for the Q as . ωhWT i Q = (22.1.11) hPDi By further letting ω = 2π/T , we lent further physical interpretation to express Q as . hWT i total energy stored Q = 2π = 2π (22.1.12) hPDiT Energy dissipated/cycle In a cavity, the energy can dissipate in either the dielectric loss or the wall loss of the cavity due to the finiteness of the conductivity. It has to be re-emphasized the Q is a low-loss concept, and hence, the above formulas are only approximately true. 22.1.2 Relation to Bandwidth and Pole Location The resonance of a system is related to the pole of the transfer function . For instance, in our previous examples of the RLC tank circuit, the admittance Y can be thought of as a transfer function in linear system theory: The input is the voltage, while the output is the current. If we encounter the resonance of the system at a particular frequency, the transfer function becomes infinite. This infinite value can be modeled by a pole of the transfer function in the complex ω plane. In other words, in the vicinity of the pole in the frequency domain, the transfer function of the system can be approximated by a single pole which is A A Y (ω) ∼ = (22.1.13) ω − ωR ω − ω0 − jα where we have assumed that ωR = ω0 + jα, the resonant frequency is complex. In principle, when ω = ωR, the reflection coefficient becomes infinite, but this does not happen in practice because ωR is complex, and ω, the operating frequency is real. In other words, the pole is displaced slightly off the real axis to account for loss. Using a single pole approximation, it is quite clear√ that |Y (ω)| would peak at ω = ω0. At ω = ω0 ± α, the magnitude of |Y (ω)| will be 1/ 2 smaller, or that the power which is proportional to |Y (ω)|2 will be half as small. Quality Factor of Cavities, Mode Orthogonality 237 Therefore, the half-power points compared to the peak are at ω = ω0 ± α. Therefore, the full-width half maximum (FWHM) bandwidth is ∆ω = 2α. Thus the Q can be written as in terms of the half-power bandwidth ∆ω of the system, viz., . Q = ω0/∆ω (22.1.14) The above implies that the narrower the bandwidth, the higher the Q of the system. Typical plots of transfer function versus frequency for a system with different Q’s are shown in Figure 22.3. Figure 22.3: A typical system response versus frequency for different Q’s using (22.1.4). The Q is altered by changing the resistor R in the circuit. 22.1.3 Wall Loss and Q for a Metallic Cavity If the cavity is filled with air, then the loss comes mainly from the metallic loss or copper-loss from the cavity wall. In this case, the power dissipated on the wall is given by [32] 1 ∗ 1 ∗ hPDi = Figure 22.4: Due to high σ in the metal, and large |βt|, phase matching condition requires the transmitted wave vector to be almost normal to the surface. For such a wave, we can approximaten ˆ × E = HtZm where Zm is the intrinsic impedance q µ q µ p ωµ for the metallic conductor, as shown in Section 8.1, which is Zm = ≈ σ = (1+ εm −j ω 2σ 3 jσ j), where we have assumed that εm ≈ − ω , and Ht is the tangential magnetic field. This relation between E and H will ensure that power is flowing into the metallic surface. Hence, r r 1 ωµ 2 1 ωµ 2 hPDi = 3When an electromagnetic wave enters a conductive region with a large β, it can be shown that the wave is refracted to propagate normally to the surface as shown in Figure 22.4, and hence, this formula can be applied. Quality Factor of Cavities, Mode Orthogonality 239 Q, a dimensionless quantity, is roughly proportional to V Q ∼ (22.1.19) Sδ where V is the volume of the cavity, while S is its surface area. From the above, it is noted that a large cavity compared to its skin depth has a larger Q than an small cavity. 22.1.4 Example: The Q of TM110 Mode For the TM110 mode, as can be seen from the previous lecture, the only electric field is E =zE ˆ z, where πx πy E = E sin sin (22.1.20) z 0 a b The magnetic field can be derived from the electric field using Maxwell’s equation or Faraday’s law, and jω ∂ j π πx πy H = E = b E sin cos (22.1.21) x ω2µ ∂y Z ωµ 0 a b −jω ∂ j π πx πy H = E = − a E cos sin (22.1.22) y ω2µ ∂x Z ωµ 0 a b Therefore ¢ ¢ ¢ ¢ 0 b a 2 h 2 2i |H| dV = dx dy dz |Hx| + |Hy| V −d 0¢ 0¢ ¢ 2 0 b a |E0| = 2 2 dx dy dz ω µ −d 0 0 π 2 πx πy π 2 πx πy sin2 cos2 + cos2 sin2 b a b a a b |E |2 π2 a b = 0 + d (22.1.23) ω2µ2 4 b a A cavity has six faces, finding the tangential exponent at each face and integrate ¢ ¢ b a h 2 2i |Ht| dS = 2 dx dy |Hx| + |Hy| 0 0 S ¢ ¢ ¢ ¢ 0 a 0 b 2 2 + 2 dx dz |Hx(y = 0)| + 2 dy dz |Hy(x = 0)| −d 0 −d 0 2 2 2 |E |2 π2ab 1 1 2 π ad 2 π bd = 0 + + b |E |2 + a |E |2 ω2µ2 4 a2 b2 ω2µ2 0 2 ω2µ2 0 2 π2 |E |2 b a ad bd = 0 + + + (22.1.24) ω2µ2 2a 2b b2 a2 240 Electromagnetic Field Theory Hence the Q is 2 ad + bd Q = b a (22.1.25) δ b a ad bd 2a + 2b + b2 + a2 The result shows that the larger the cavity, the higher the Q. This is because the Q, as mentioned before, is the ratio of the energy stored in a volume to the energy dissipated over the surface of the cavity. 22.2 Mode Orthogonality and Matrix Eigenvalue Prob- lem It turns out that the modes of a waveguide or a resonator are orthogonal to each other. This is intimately related to the orthogonality of eigenvectors of a matrix operator.4 Thus, it is best to understand this by the homomorphism between the electromagnetic mode problem and the matrix eigenvalue problem. Because of this similarity, electromagnetic modes are also called eigenmodes. Thus it is prudent that we revisit the matrix eigenvalue problem (EVP) here. 22.2.1 Matrix Eigenvalue Problem (EVP) It is known in matrix theory that if a matrix is hermitian, then its eigenvalues are all real. Furthermore, their eigenvectors with distinct eigenvalues are orthogonal to each other [77]. Assume that an eigenvalue and an eigenvector exists for the hermitian matrix A. Then A · vi = λivi (22.2.1) † Dot multiplying the above from the left by vi where † indicates conjugate transpose, then the above becomes † † vi · A · vi = λivi · vi (22.2.2) † † Since A is hermitian, or A = A, then the quantity vi · A · vi is purely real. Moreover, the † quantity vi · vi is positive real. So in order for the above to be satisfied, λi has to be real. To prove orthogonality of eigenvectors, now, assume that A has two eigenvectors with distinct eigenvalues such that A · vi = λivi (22.2.3) A · vj = λjvj (22.2.4) † † Left dot multiply the first equation with vj and do the same to the second equation with vi , one gets † † vj · A · vi = λivj · vi (22.2.5) † † vi · A · vj = λjvi · vj (22.2.6) 4This mathematical homomorphism is not discussed in any other electromagnetic textbooks. Quality Factor of Cavities, Mode Orthogonality 241 Taking the conjugate transpose of (22.2.5) in the above, and since A is hermitian, their left- hand sides (22.2.5) and (22.2.6) become the same. Subtracting the two equations, we arrive at † 0 = (λi − λj)vj · vi (22.2.7) For distinct eigenvalues, λi 6= λj, the only way for the above to be satisfied is that † vj · vi = Ciδij (22.2.8) Hence, eigenvectors of a hermitian matrix with distinct eigenvalues are orthogonal to each other. The eigenvalues are also real. 22.2.2 Homomorphism with the Waveguide Mode Problem We shall next show that the problem for finding the waveguide modes or eigenmodes is analogous to the matrix eigenvalue problem. The governing equation for a waveguide mode is a BVP involving the reduced wave equation previously derived, or 2 2 2 2 ∇sψi(rs) + βisψi(rs) = 0, ⇒ −∇sψi(rs) = βisψi(rs) (22.2.9) with the pertinent homogeneous Dirichlet or Neumann boundary condition, depending on if 2 TE or TM modes are considered. In the above, the differential operator ∇s is analogous to 2 the matrix operator A, the eigenfunction ψi(rs) is analogous to the eigenvector vi, and βis is analogous to the eigenvalue λi. 2 In the above, we can think of ∇s as a linear operator that maps a function to another function, viz., 2 ∇sψi(rs) = fi(rs) (22.2.10) Hence it is analogous to a matrix operator A, that maps a vector to another vector, viz., A · x = f (22.2.11) We shall elaborate this further next. Discussion on Functional Space 5 To think of a function ψ(rs) as a vector, one has to think in the discrete world. If one needs to display the function ψ(rs), on a computer, one will evaluate the function ψ(rs) at discrete N locations rls, where l = 1, 2, 3,...,N. For every rls or every l, there is a scalar number ψ(rls). These scalar numbers can be stored in a column vector in a computer indexed by l. The larger N is, the better is the discrete approximation of ψ(rs). In theory, N needs to be infinite to describe this function exactly.6 5Some of these concepts are discussed in [35, 131]. 6 In mathematical parlance, the index for ψ(rs) is uncountably infinite or nondenumerable. 242 Electromagnetic Field Theory From the above discussion, a function is analogous to a vector and a functional space is analogous to a vector space in linear or matrix algebra. However, a functional space is infinite dimensional. But in order to compute on a computer with finite resource, such functions are approximated with finite length vectors. Or infinite dimensional vector spaces are replaced with finite dimensional ones to make the problem computable. Such an infinite dimensional functional space is also called a Hilbert space. It is also necessary to define the inner product between two vectors in a functional space just as inner product between two vectors in an matrix vector space. The inner product (or dot product) between two vectors in matrix vector space is N t X vi · vj = vi,lvj,l (22.2.12) l=1 The analogous inner product between two vectors in function space is7 ¢ hψi, ψji = drsψi(rs)ψj(rs) (22.2.13) S where S denotes the cross-sectional area of the waveguide over which the integration is per- formed. The left-hand side is the shorthand notation for inner product in functional space or the infinite dimensional functional Hilbert space. Another requirement for a vector in a functional Hilbert space is that it contains finite energy or that ¢ 2 Ef = drs|ψi(rs)| (22.2.14) S is finite. The above is analogous to that for a matrix vector v as N X 2 Em = |vl| (22.2.15) l=1 The square root of the above is often used to denote the “length”, the “metric”, or the “norm” of the vector. Finite energy also implies that the vectors are of finite length. 22.2.3 Proof of Orthogonality of Waveguide Modes8 Because of the aforementioned discussion, we see the similarity between a functional Hilbert space, and the matrix vector space. In order to use the result of the matrix EVP, one key 2 step is to prove that the operator ∇s is hermitian. In matrix algebra, if the elements of a matrix is explicitly available, a matrix A is hermitian if ∗ Aij = Aji (22.2.16) 7 In many math books, the conjugation of the first function ψi is implied, but here, we follow the electro- magnetic convention that the conjugation of ψi is not implied unless explicitly stated. 8This may be skipped on first reading. Quality Factor of Cavities, Mode Orthogonality 243 But if a matrix element is not explicitly available, a different way to define hermiticity is required: we will use inner products to define it. A matrix operator is hermitian if † † ∗ † † † † † xi · A · xj = xj · A · xi = xj · A · xi = xj · A · xi (22.2.17) The first equality follows from standard matrix algebra,9 the second equality follows if A = † A , or that A is hermitian. The last equality follows because the quantity in the parenthesis is a scalar, and hence, its conjugate transpose is just its conjugate. Therefore, if a matrix A is hermitian, then ∗ † † xi · A · xj = xj · A · xi (22.2.18) The above inner product can be used to define the hermiticity of the matrix operator A. It can be easily extended to define the hermiticity of an operator in an infinite dimensional Hilbert space where the matrix elements are not explicitly available. Hence, using the inner product definition in (22.2.13) for infinite dimensional functional 2 Hilbert space, a functional operator ∇s is hermitian if ∗ 2 ∗ 2 ∗ hψi , ∇sψji = (hψj , ∇sψii) (22.2.19) We can rewrite the left-hand side of the above more explicitly as ¢ ∗ 2 ∗ 2 hψi , ∇sψji = drsψi (rs)∇sψj(rs) (22.2.20) S and then the right-hand side of (22.2.19) can be rewritten more explicitly as ¢ ∗ 2 ∗ 2 ∗ (hψj , ∇sψii) = drsψj(rs)∇sψi (rs) (22.2.21) S To prove the above equality in (22.2.19), one uses the identity that ∗ ∗ 2 ∗ ∇s · [ψi (rs)∇sψj(rs)] = ψi (rs)∇sψj(rs) + ∇sψi (rs) · ∇sψj(rs) (22.2.22) Integrating the above over the cross sectional area S, and invoking Gauss divergence theorem in 2D, one gets that ¢ ¢ ∗ ∗ 2 dlnˆ · (ψi (rs)∇sψj(rs)) = drs ψi (rs)∇sψj(rs) C S ¢ ∗ + drs (∇ψi (rs) · ∇sψj(rs)) (22.2.23) S where C the the contour bounding S or the waveguide wall. By applying the boundary condition that ψi(rs) = 0 or thatn ˆ · ∇sψj(rs) = 0, or a mixture thereof, then the left-hand side of the above is zero. This will be the case be it TE or TM modes. ¢ ¢ ∗ 2 ∗ 0 = drs ψi (rs)∇sψj(rs) + drs (∇ψi (rs) · ∇sψj(rs)) (22.2.24) S S † † † 9(A · B · C)† = C · B · A [77]. 244 Electromagnetic Field Theory Applying the same treatment to (22.2.21), we get ¢ ¢ 2 ∗ ∗ 0 = drs ψj(rs)∇sψi (rs) + drs (∇ψi (rs) · ∇sψj(rs)) (22.2.25) S S The above two equations indicate that ∗ 2 ∗ 2 ∗ hψi , ∇sψji = (hψj , ∇sψii) (22.2.26) 2 proving that the operator ∇s is hermitian. One can then use the above property to prove the orthogonality of the eigenmodes when they have distinct eigenvalues, the same way we have proved the orthogonality of eigenvectors. The above proof can be extended to the case of a resonant cavity. The orthogonality of resonant cavity modes is also analogous to the orthogonality of eigenvectors of a hermitian operator. Lecture 23 Scalar and Vector Potentials 23.1 Scalar and Vector Potentials for Time-Harmonic Fields Now that we have studied the guidance of waves by waveguides, and the trapping of elec- tromagnetic waves by cavity resonator, it will be interesting to consider how electromagnetic waves radiate from sources. This is best done via the scalar and vector potential formulation. Previously, we have studied the use of scalar potential Φ for electrostatic problems. Then we learnt the use of vector potential A for magnetostatic problems. Now, we will study the combined use of scalar and vector potential for solving time-harmonic (electrodynamic) problems. This is important for bridging the gap between static regime where the frequency is zero or low, and dynamic regime where the frequency is not low. For the dynamic regime, it is important to understand the radiation of electromagnetic fields. Electrodynamic regime is important for studying antennas, communications, sensing, wireless power transfer ap- plications, and many more. Hence, it is imperative that we understand how time-varying electromagnetic fields radiate from sources. It is also important to understand when static or circuit (quasi-static) regimes are im- portant. The circuit regime solves problems that have fueled the microchip and integrated circuit design (ICD) industry, and it is hence imperative to understand when electromagnetic problems can be approximated with simple circuit problems and solved using simple laws such as Kirchhoff current law (KCL) an Kirchhoff voltage law (KVL). 245 246 Electromagnetic Field Theory Figure 23.1: Plot of the dynamic electric field around a dipole source, where only the field in the upper half-space is shown. Close to the source, the field ressembles that of a static electric dipole, but far away from the source, the electromagnetic field is detached from the source: the source starts to shed energy to the far region. 23.2 Scalar and Vector Potentials for Statics, A Review Previously, we have studied scalar and vector potentials for electrostatics and magnetostatics where the frequency ω is identically zero. The four Maxwell’s equations for a homogeneous medium are then ∇ × E = 0 (23.2.1) ∇ × H = J (23.2.2) ∇ · εE = % (23.2.3) ∇ · µH = 0 (23.2.4) Looking at the first equation above, and using the knowledge that ∇ × (∇Φ) = 0, we can construct a solution to (23.2.1) easily. Thus, in order to satisfy the first of Maxwell’s equations or Faraday’s law above, we let E = −∇Φ (23.2.5) Using the above in (23.2.3), we get, ∇ · ε∇Φ = −% (23.2.6) Then for a homogeneous medium where ε is a constant, ∇ · ε∇Φ = ε∇ · ∇Φ = ε∇2Φ, and we have % ∇2Φ = − (23.2.7) ε which is the Poisson’s equation for electrostatics. Scalar and Vector Potentials 247 Now looking at (23.2.4) where ∇ · µH = 0, we let µH = ∇ × A (23.2.8) Since ∇ · (∇ × A) = 0, the last of Maxwell’s equations (23.2.4) is automatically satisfied. Next, using the above in the second of Maxwell’s equations above, we get ∇ × ∇ × A = µJ (23.2.9) Now, using the fact that ∇ × ∇ × A = ∇(∇ · A) − ∇2A, and Coulomb gauge that ∇ · A = 0, we arrive at ∇2A = −µJ (23.2.10) which is the vector Poisson’s equation. Next, we will repeat the above derivation when ω 6= 0. 23.2.1 Scalar and Vector Potentials for Electrodynamics To this end, assuming linearity, we will start with frequency domain Maxwell’s equations with sources J and % included, and later see how these sources J and % can radiate electromagnetic fields. Maxwell’s equations in the frequency domain are ∇ × E = −jωµH (23.2.11) ∇ × H = jωεE + J (23.2.12) ∇ · εE = % (23.2.13) ∇ · µH = 0 (23.2.14) In order to satisfy the last Maxwell’s equation, as before, we let µH = ∇ × A (23.2.15) Now, using (23.2.15) in (23.2.11), we have ∇ × (E + jωA) = 0 (23.2.16) Since ∇ × (∇Φ) = 0, the above implies that E + jωA = −∇Φ, or E = −jωA − ∇Φ (23.2.17) The above implies that the electrostatic theory of letting E = −∇Φ we have learnt previously in Section 3.3.1 is not exactly correct when ω 6= 0. The second term above, in accordance to Faraday’s law, is the contribution to the electric field from the time-varying magnetic field, and hence, is termed the induction term.1 Furthermore, the above shows that once A and Φ are known, one can determine the fields H and E assuming that J and % are given. To this end, we will derive equations for A and Φ in terms of the sources J and %. Substituting (23.2.15) and (23.2.17) into (23.2.12) gives ∇ × ∇ × A = −jωµε(−jωA − ∇Φ) + µJ (23.2.18) 1Notice that in electrical engineering, most concepts related to magnetic fields are inductive! 248 Electromagnetic Field Theory Or upon rearrangement, after using that ∇ × ∇ × A = ∇∇ · A − ∇ · ∇A, we have ∇2A + ω2µεA = −µJ + jωµε∇Φ + ∇∇ · A (23.2.19) Moreover, using (23.2.17) in (23.2.14), we have % ∇ · (jωA + ∇Φ) = − (23.2.20) ε In the above, (23.2.19) and (23.2.20) represent two equations for the two unknowns A and Φ, expressed in terms of the known quantities, the sources J and %. But these equations are coupled to each other. They look complicated and are rather unwieldy to solve at this point. Fortunately, the above can be simplified! As in the magnetostatic case, the vector potential A is not unique. To show this, one can always construct a new A0 = A + ∇Ψ that produces the same magnetic field µH via (23.2.8), since ∇ × (∇Ψ) = 0. It is quite clear that µH = ∇ × A = ∇ × A0. Moreover, one can further show that Φ is also non-unique [48]. Namely, with A0 = A + ∇Ψ (23.2.21) Φ0 = Φ − jωΨ (23.2.22) it can be shown that the new A0 and Φ0 produce the same E and H field. The above is known as gauge transformation, clearly showing the non-uniqueness of A and Φ. To make them unique, in addition to specifying what ∇ × A should be in (23.2.15), we need to specify its divergence or ∇ · A as in the electrostatic case.2 A clever way to specify the divergence of A is to choose it to simplify the complicated equations above in (23.2.19). We choose a gauge so that the last two terms in the equation (23.2.19) cancel each other. In other words, we let ∇ · A = −jωµεΦ (23.2.23) The above is judiciously chosen so that the pertinent equations (23.2.19) and (23.2.20) will be simplified and decoupled. With the use of (23.2.23) in (23.2.19) and (23.2.20), they now become ∇2A + ω2µεA = −µJ (23.2.24) % ∇2Φ + ω2µεΦ = − (23.2.25) ε Equation (23.2.23) is known as the Lorenz gauge3 and the above equations are Helmholtz equations with source terms. Not only are these equations simplified, they can be solved independently of each other since they are decoupled from each other. Equations (23.2.24) and (23.2.25) can be solved using the Green’s function method we have learnt previously. Equation (23.2.24) actually constitutes three scalar equations for the three x, y, z components, namely that 2 2 ∇ Ai + ω µεAi = −µJi (23.2.26) 2This is akin to that given a vector A, and an arbitrary vector k, in addition to specifying what k × A is, it is also necessary to specify what k · A is to uniquely specify A. 3Please note that this Lorenz is not the same as Lorentz. Scalar and Vector Potentials 249 where i above can be x, y, or z. Therefore, (23.2.24) and (23.2.25) together constitute four scalar equations similar to each other. Hence, we need only to solve their point-source response, or the Green’s function of these equations by solving ∇2g(r, r0) + β2g(r, r0) = −δ(r − r0) (23.2.27) where β2 = ω2µε. Previously, we have shown that when β = 0, 1 g(r, r0) = g(|r − r0|) = 4π|r − r0| When β 6= 0, the correct solution is 0 e−jβ|r−r | g(r, r0) = g(|r − r0|) = (23.2.28) 4π|r − r0| which can be verified by back substitution. By using the principle of linear superposition, or convolution, the solutions to (23.2.24) and (23.2.25) are then ¦ ¦ 0 e−jβ|r−r | A(r) = µ dr0J(r0)g(|r − r0|) = µ dr0J(r0) (23.2.29) 4π|r − r0| ¦ ¦ 0 1 1 e−jβ|r−r | Φ(r) = dr0%(r0)g(|r − r0|) = dr0%(r0) (23.2.30) ε ε 4π|r − r0| In the above dr0 is the shorthand notation for dx dy dz and hence, they are still volume integrals. The above are three-dimensional convolutional integrals in space. 23.2.2 More on Scalar and Vector Potentials It is to be noted that Maxwell’s equations are symmetrical and this is especially so when we 4 add a magnetic current M to Maxwell’s equations and magnetic charge %m to Gauss’s law. Thus the equations then become ∇ × E = −jωµH − M (23.2.31) ∇ × H = jωεE + J (23.2.32) ∇ · µH = %m (23.2.33) ∇ · εE = % (23.2.34) The above can be solved in two stages, using the principle of linear superposition. First, we can set M = 0, %m = 0, and J 6= 0, % 6= 0, and solve for the fields as we have done. Second, we can set J = 0, % = 0 and M 6= 0, %m 6= 0 and solve for the fields next. Then the total general solution, by linearity, is just the linear superposition of these two solutions. 4In fact, Maxwell exploited this symmetry [38]. 250 Electromagnetic Field Theory For the second case, we can define an electric vector potential F such that D = −∇ × F (23.2.35) and a magnetic scalar potential Φm such that H = −∇Φm − jωF (23.2.36) By invoking duality principle, one gather that [47] ¦ ¦ 0 e−jβ|r−r | F(r) = ε dr0M(r0)g(|r − r0|) = ε dr0M(r0) (23.2.37) 4π|r − r0| ¦ ¦ 0 1 1 e−jβ|r−r | Φ (r) = dr0% (r0)g(|r − r0|) = dr0% (r0) (23.2.38) m µ m µ m 4π|r − r0| As mentioned before, even though magnetic sources do not exist, they can be engineered. In many engineering designs, one can use fictitious magnetic sources to enrich the diversity of electromagnetic technologies. 23.3 When is Static Electromagnetic Theory Valid? We have learnt in the previous section that for electrodynamics, E = −∇Φ − jωA (23.3.1) where the second term above on the right-hand side is due to induction, or the contribution to the electric field from the time-varying magnetic filed. Hence, much things we learn in potential theory that E = −∇Φ is not exactly valid. But simple potential theory that E = −∇Φ is very useful because of its simplicity. We will study when static electromagnetic theory can be used to model electromagnetic systems. Since the third and the fourth Maxwell’s equations are derivable from the first two, let us first study when we can ignore the time derivative terms in the first two of Maxwell’s equations, which, in the frequency domain, are ∇ × E = −jωµH (23.3.2) ∇ × H = jωεE + J (23.3.3) When the terms multiplied by jω above, which are associated with time derivatives, can be ignored, then electrodynamics can be replaced with static electromagnetics, which are much simpler.5 Quasi-static electromagnetic theory eventually gave rise to circuit theory and telegraphy technology. Circuit theory consists of elements like resistors, capacitors, and inductors. Given that we have now seen electromagnetic theory in its full form, we would like to ponder when we can use simple static electromagnetics to describe electromagnetic phenomena. 5That is why Ampere’s law, Coulomb’s law, and Gauss’ law were discovered first. Scalar and Vector Potentials 251 Figure 23.2: The electric and magnetic fields are great contortionists around a perfectly conducting particle. They deform themselves to satisfy the boundary conditions,n ˆ × E = 0, andn ˆ · H = 0 on the PEC surface, even when the particle is very small. In other words, the fields vary on the length-scale of 1/L. Hence, ∇ ∼ 1/L which is large when L is small. To see this lucidly, it is best to write Maxwell’s equations in dimensionless units or the same units. Say if we want to solve Maxwell’s equations for the fields close to an object of size L as shown in Figure 23.2. This object can be a small particle like the sphere, or it could be a capacitor, or an inductor, which are small; but how small should it be before we can apply static electromagnetics? It is clear that these E and H fields will have to satisfy boundary conditions,n ˆ × E = 0, andn ˆ · H = 0 on the PEC surface, which is de rigueur in the vicinity of the object as shown in Figure 23.2 even when the frequency is low or the wavelength long. The fields become great contortionists in order to do so. Hence, we do not expect a constant field around the object but that the field will vary on the length scale of L. So we renormalize our length scale by this length L by defining a new dimensionless coordinate system such that6 x y z x0 = , y0 = , z0 = (23.3.4) L L L In other words, by so doing, then Ldx0 = dx, Ldy0 = dy, and Ldz0 = dz, and ∂ 1 ∂ ∂ 1 ∂ ∂ 1 ∂ = , = , = (23.3.5) ∂x L ∂x0 ∂y L ∂y0 ∂z L ∂z0 Take for example a function f(x) = sin(πx/L) which is a periodic function varying on the length scale of L having a period of L. Then it is quite clear that df(x)/dx = (π/L) cos(πx/L). When L is small representing a rapidly varying function, df(x)/dx ∼ O(1/L).7 But in the new variable, f(Lx0) = sin(πx0) and df/dx0 = cos(πx0) which is O(1). 6This dimensional analysis is often used by fluid dynamicists to study fluid flow problems [132]. 7Reads order of 1/L of order 1/L. 252 Electromagnetic Field Theory 1 0 0 In this manner, ∇ = L ∇ where ∇ is dimensionless; or ∇ will be very large when it operates on fields that vary on the length scale of very small L, where ∇0, which is an O(1) operator, will not be large because it is in coordinates normalized with respect to L. Then, the first two of Maxwell’s equations become 1 ∇0 × E = −jωµ H (23.3.6) L 0 1 ∇0 × H = jωε E + J (23.3.7) L 0 Here, we still have apples and oranges to compare with since E and H have different units; we cannot compare quantities if they have different units. For instance, the ratio of E to the H field has a dimension of impedance. To bring them to the same unit, we define a new E0 such that 0 η0E = E (23.3.8) p ∼ 0 where η0 = µ0/ε0 = 377 ohms in vacuum. In this manner, the new E has the same unit as the H field. Then, (23.3.6) and (23.3.7) become η 0 ∇0 × E0 = −jωµ H (23.3.9) L 0 1 ∇0 × H = jωε η E0 + J (23.3.10) L 0 0 With this change, the above can be rearranged to become 0 0 L ∇ × E = −jωµ0 H (23.3.11) η0 0 0 ∇ × H = jωε0η0LE + LJ (23.3.12) p By letting η0 = µ0/ε0, the above can be further simplified to become ω ∇0 × E0 = −j LH (23.3.13) c0 ω ∇0 × H = j LE0 + LJ (23.3.14) c0 Notice now that in the above, H, E0, and LJ have the same unit, and ∇0 is dimensionless and is of order one, and ωL/c0 is also dimensionless. Therefore, in the above, one can compare terms, and ignore the frequency dependent jω term when ω L 1 (23.3.15) c0 The above is the same as kL 1 where k = ω/c0 = 2π/λ0. Thus, when L 2π 1 (23.3.16) λ0 Scalar and Vector Potentials 253 the jω terms can be ignored and the first two Maxwell’s equations become static equations. Consequently, the above criteria are for the validity of the static approximation when the time- derivative terms in Maxwell’s equations can be ignored. When these criteria are satisfied, then Maxwell’s equations can be simplified to and approximated by the following equations ∇0 × E0 = 0 (23.3.17) ∇0 × H = LJ (23.3.18) which are the static equations, Faraday’s law and Ampere’s law of electromagnetic theory. They can be solved together with Gauss’ laws, or the third or the fourth Maxwell’s equations. In other words, one can solve, even in optics, where ω is humongous or the wavelength very short, using static analysis if the size of the object L is much smaller than the optical wavelength which is about 400 nm for blue light. Nowadays, plasmonic nano-particles of about 10 nm can be made. If the particle is small enough compared to wavelength of the light, electrostatic analysis can be used to study their interaction with light. And hence, static electromagnetic theory can be used to analyze the wave-particle interaction. This was done in one of the homeworks. Figure 23.3 shows an incident light whose wavelength is much longer than the size of the particle. The incident field induces an electric dipole moment on the particle, whose external field can be written as a3 E = (ˆr2 cos θ + θˆsin θ) E (23.3.19) s r s while the incident field E0 and the interior field Ei to the particle can be expressed as ˆ E0 =zE ˆ 0 = (ˆr cos θ − θ sin θ)E0 (23.3.20) ˆ Ei =zE ˆ i = (ˆr cos θ − θ sin θ)Ei (23.3.21) By matching boundary conditions, as was done in the homework, it can be shown that εs − ε Es = E0 (23.3.22) εs + 2ε 3ε Ei = E0 (23.3.23) εs + 2ε For a plasmonic nano-particle, the particle medium behaves like a plasma (see Lecture 8), and εs in the above can be negative, making the denominators of the above expression very close to zero. Therefore, the amplitude of the internal and scattered fields can be very large when this happens, and the nano-particles will glitter in the presence of light. Even the ancient Romans realized this! Figure 23.4 shows a nano-particle induced in plasmonic oscillation by a light wave. Figure 23.5 shows that different color fluids can be obtained by immersing nano-particles in fluids with different background permittivity (ε in (23.3.22) and (23.3.23)) causing the plasmonic particles to resonate at different frequencies. 254 Electromagnetic Field Theory Figure 23.3: A plane electromagnetic wave incident on a particle. When the particle size is small compared to wavelength, static analysis can be used to solve this problem (courtesy of Kong [32]). Figure 23.4: A nano-particle undergoing electromagnetic oscillation when an electromagnetic wave impinges on it (courtesy of sigmaaldric.com). The oscillation is inordinately large when the incident wave’s frequency coincides with the resonance frequency of the plasmonic particle. Scalar and Vector Potentials 255 Figure 23.5: Different color fluids containing nano-particles can be obtained by changing the permittivity of the background fluids (courtesy of nanocomposix.com). In (23.3.16), this criterion has been expressed in terms of the dimension of the object L compared to the wavelength λ0. Alternatively, we can express this criterion in terms of transit time. The transit time for an electromagnetic wave to traverse an object of size L is τ = L/c0 and ω = 2π/T where T is the period of one time-harmonic oscillation. Therefore, (23.3.15) can be re-expressed as 2πτ 1 (23.3.24) T The above implies that if the transit time τ needed for the electromagnetic field to traverse the object of length L is much small than the period of oscillation of the electromagnetic field, then static theory can be used. The finite speed of light gives rise to delay or retardation of electromagnetic signal when it propagates through space. When this retardation effect can be ignored, then static elec- tromagnetic theory can be used. In other words, if the speed of light had been infinite, then there would be no retardation effect, and static electromagnetic theory could always be used. Alternatively, the infinite speed of light will give rise to infinite wavelength, and criterion (23.3.16) will always be satisfied, and static theory prevails always. 23.3.1 Quasi-Static Electromagnetic Theory In closing, we would like to make one more remark. The right-hand side of (23.3.11), which is Faraday’s law, is essential for capturing the physical mechanism of an inductor and flux linkage. And yet, if we drop it, there will be no inductor in this world. To understand this dilemma, let us rewrite (23.3.11) in integral form, namely, ¤ 0 L E · dl = −jωµ0 dS · H (23.3.25) C η0 S In the inductor, the right-hand side has been amplified by multiple turns, effectively increasing S, the flux linkage area. Or one can think of an inductor as having a much longer effective length Leff when untwined so as to compensate for decreasing frequency ω. Hence, the importance of flux linkage or the inductor in circuit theory is not diminished unless ω = 0. 256 Electromagnetic Field Theory By the same token, displacement current in (23.3.11) can be enlarged by using capacitors. In this case, even when no electric current J flows through the capacitor, displacement current flows and the generalized Ampere’s law becomes ¤ 0 H · dl = jωεη0L dS · E (23.3.26) C S The displacement in a capacitor cannot be ignored unless ω = 0. Therefore, when ω 6= 0, or in quasi-static case, inductors and capacitors in circuit theory are important, because they amplify the flux linkage and the displacement current effects, as we shall study next. In summary, the full physics of Maxwell’s equations is not lost in circuit theory: the induction term in Faraday’s law, and the displacement current in Ampere’s law are still retained. That explains the success of circuit theory in electromagnetic engineering! Lecture 24 Circuit Theory Revisited Circuit theory is one of the most successful and often used theories in electrical engineering. Its success is mainly due to its simplicity: it can capture the physics of highly complex circuits and structures, which is very important in the computer and micro-chip industry (or the IC design industry). Simplicity rules! Now, having understood electromagnetic theory in its full glory, it is prudent to revisit circuit theory and study its relationship to electromagnetic theory [31, 32,54, 65]. The two most important laws in circuit theory are Kirchoff current law (KCL) and Kirch- hoff voltage law (KVL) [14, 52]. These two laws are derivable from the current continuity equation and from Faraday’s law. 24.1 Kirchhoff Current Law Figure 24.1: Schematic showing the derivation of Kirchhoff current law. All currents flowing into a node must add up to zero. 257 258 Electromagnetic Field Theory Kirchhoff current law (KCL) is a consequence of the current continuity equation, or that ∇ · J = −jω% (24.1.1) It is a consequence of charge conservation. But it is also derivable from generalized Ampere’s law and Gauss’ law for charge.1 First, we assume that all currents are flowing into a node as shown in Figure 24.1, and that the node is non-charge accumulating with ω → 0. Then the charge continuity equation becomes2 ∇ · J = 0 (24.1.2) By integrating the above current continuity equation over a volume containing the node, it is easy to show that N X Ii = 0 (24.1.3) i which is the statement of KCL. This is shown for the schematic of Figure 24.1. 24.2 Kirchhoff Voltage Law Kirchhoff voltage law is the consequence of Faraday’s law. For the truly static case when ω = 0, it is ∇ × E = 0 (24.2.1) The above implies that E = −∇Φ, from which we can deduce that − E · dl = 0 (24.2.2) C For statics, the statement that E = −∇Φ also implies that we can define a voltage drop between two points, a and b to be ¢ ¢ b b Vba = − E · dl = ∇Φ · dl = Φ(rb) − Φ(ra) = Vb − Va (24.2.3) a a ¡ b The equality a ∇Φ · dl = Φ(rb) − Φ(ra) can be understood by expressing this integral in one dimension along a straight line segment, or that ¢ b d Φ · dx = Φ(rb) − Φ(ra) (24.2.4) a dx 1Some authors will say that charge conservation is more fundamental, and that Gauss’ law and Ampere’s law are consistent with charge conservation and the current continuity equation. 2One can also say that this is a consequenc of static Ampere’s law, ∇ × H = J. By taking the divergence of this equation yields (24.1.2) directly. Circuit Theory Revisited 259 A curved line can be thought of as a concatenation of many small straight line segments. As has been shown before, to be exact, E = −∇Φ − ∂/∂tA, but we have ignored the induction effect. Therefore, this concept is only valid in the low frequency or long wavelength limit, or that the dimension over which the above is applied is very small so that retardation effect can be ignored. A good way to remember the above formula is that if Vb > Va. Since E = −∇Φ, then the electric field points from point a to point b: Electric field always points from the point of higher potential to point of lower potential. Faraday’s law when applied to the static case for a closed loop of resistors shown in Figure 24.2 gives Kirchhoff voltage law (KVL), or that N X Vj = 0 (24.2.5) i Notice that the voltage drop across a resistor is always positive, since the voltages to the left of the resistors in Figure 24.2 are always higher than the voltages to the right of the resistors. This implies that internal to the resistor, there is always an electric field that points from the left to the right. If one of the voltage drops is due to a voltage source, it can be modeled by a negative resistor as shown in Figure 24.3. The voltage drop across a negative resistor is opposite to that of a positive resistor. As we have learn from the Poynting’s theorem, negative resistor gives out energy instead of dissipates energy. Figure 24.2: Kichhoff voltage law where the sum of all voltages around a loop is zero, which is the consequence of static Faraday’s law. 260 Electromagnetic Field Theory Figure 24.3: A voltage source can also be modeled by a negative resistor. Faraday’s law for the time-varying B flux case is ∂B ∇ × E = − (24.2.6) ∂t Writing the above in integral form, one gets ¢ d − E · dl = B · dS (24.2.7) C dt s We can apply the above to a loop shown in Figure 24.4, or a loop C that goes from a to b to c to d to a. We can further assume that this loop is very small compared to wavelength so that potential theory that E = −∇Φ can be applied. Furthermore, we assume that this loop C does not have any magnetic flux through it so that the right-hand side of the above can be set to zero, or − E · dl = 0 (24.2.8) C Circuit Theory Revisited 261 Figure 24.4: The Kirchhoff voltage law for a circuit loop consisting of resistor, inductor, and capacitor can also be derived from Faraday’s law at low frequency (courtesy of Ramo et al). Figure 24.5: The voltage-current relation of an inductor can be obtained by unwrapping an inductor coil, and then calculating its flux linkage. Notice that this loop does not go through the inductor, but goes directly from c to d. Then there is no flux linkage in this loop and thus ¢ ¢ ¢ ¢ b c d a − E · dl − E · dl − E · dl − E · dl = 0 (24.2.9) a b c d Inside the source or the battery, it is assumed that the electric field points opposite to the direction of integration dl, and hence the first term on the left-hand side of the above is positive and equal to V0(t), while the other terms are negative. Writing out the above more explicitly, after using (24.2.3), we have V0(t) + Vcb + Vdc + Vad = 0 (24.2.10) Notice that in the above, in accordance to (24.2.3), Vb > Vc, Vc > Vd, and Va > Va. Therefore, Vcb, Vdc, and Vad are all negative quantities but V0(t) > 0. We will study the contributions to each of the terms, the inductor, the capacitor, and the resistor more carefully next. 262 Electromagnetic Field Theory 24.3 Inductor To find the voltage current relation of an inductor, we apply Faraday’s law to a closed loop C0 formed by dc and the inductor coil shown in the Figure 24.5 where we have unwrapped the solenoid into a larger loop. Assume that the inductor is made of a perfect conductor, so that the electric field E in the wire is zero. Then the only contribution to the left-hand side of Faraday’s law is the integration from point d to point c, the only place in the loop C0 where E is not zero. We assume that outside the loop in the region between c and d, potential theory applies, and hence, E = −∇Φ. Now, we can connect Vdc in the previous equation to the flux linkage to the inductor. When the voltage source attempts to drive an electric current into the loop, Lenz’s law (1834)3 comes into effect, essentially, generating an opposing voltage. The opposing voltage gives rise to charge accumulation at d and c, and therefore, a low frequency electric field at the gap at dc. To this end, we form a new C0 that goes from d to c, and then continue onto the wire that leads to the inductor. But this new loop will contain the flux B generated by the inductor current. Thus ¢ ¢ c d E · dl = E · dl = −Vdc = − B · dS (24.3.1) C0 d dt S0 As mentioned before, since the wire¡ is a PEC, the integration around the loop C0 is only nonzero from d to c. In the above, S0 B · dS is the flux linkage. The inductance L is defined as the flux linkage per unit current, or ¢ L = B · dS /I (24.3.2) S0 So the voltage in (24.3.1) is then d dI V = (LI) = L (24.3.3) dc dt dt since L is time independent. Had there been a finite resistance in the wire of the inductor, then the electric field is non-zero inside the wire. Taking this into account, we have ¢ d E · dl = RLI − Vdc = − B · dS (24.3.4) dt S Consequently, dI V = R I + L (24.3.5) dc L dt Thus, to account for the loss of the coil, we add a resistor in the equation. The above becomes simpler in the frequency domain, namely Vdc = RLI + jωLI (24.3.6) 3Lenz’s law can also be explained from Faraday’s law (1831). Circuit Theory Revisited 263 24.4 Capacitance The capacitance is the proportionality constant between the charge Q stored in the capacitor, and the voltage V applied across the capacitor, or Q = CV . Then Q C = (24.4.1) V From the current continuity equation, one can easily show that in Figure 24.6, dQ d dV I = = (CV ) = C da (24.4.2) dt dt da dt where C is time independent. Integrating the above equation, one gets ¢ t 1 0 Vda(t) = Idt (24.4.3) C −∞ The above looks quite cumbersome in the time domain, but in the frequency domain, it becomes I = jωCVda (24.4.4) Figure 24.6: Schematic showing the calculation of the capacitance of a capacitor. 24.5 Resistor The electric field is not zero inside the resistor as electric field is needed to push electrons through it. As discussed in Section 8.3, a resistor is a medium where collision dominates. As is well known, J = σE (24.5.1) 264 Electromagnetic Field Theory where σ is the conductivity of the medium. From this, we deduce that Vcb = Vc − Vb is a negative number given by ¢ ¢ c c J Vcb = − E · dl = − · dl (24.5.2) b b σ where we assume a uniform current J = ˆlI/A in the resistor where ˆl is a unit vector pointing in the direction of current flow in the resistor. We can assumed that I is a constant along the length of the resistor, and thus, J · dl = Idl/A, implying that ¢ ¢ c Idl c dl Vcb = − = −I = −IR (24.5.3) b σA b σA where4 ¢ ¢ c dl c ρdl R = = (24.5.4) b σA b A Again, for simplicity, we assume long wavelength or low frequency in the above derivation. 24.6 Some Remarks In this course, we have learnt that given the sources % and J of an electromagnetic system, one can find Φ and A, from which we can find E and H. This is even true at DC or statics. We have also looked at the definition of inductor L and capacitor C. But clever engineering is driven by heuristics: it is better, at times, to look at inductors and capacitors as energy storage devices, rather than flux linkage and charge storage devices. Another important remark is that even though circuit theory is simpler that Maxwell’s equations in its full glory, not all the physics is lost in it. The physics of the induction term in Faraday’s law and the displacement current term in generalized Ampere’s law are still retained and amplified by capacitor and inductor, respectively. In fact, wave physics is still retained in circuit theory: one can make slow wave structure out a series of inductors and capacitors. The lumped-element model of a transmission line is an example of a slow-wave structure design. Since the wave is slow, it has a smaller wavelength, and resonators can be made smaller: We see this in the LC tank circuit which is a much smaller resonator in wavelength with L/λ 1 compared to a microwave cavity resonator for instance. Therefore, circuit design is great for miniaturization. The short coming is that inductors and capacitors generally have higher losses than air or vacuum. 24.7 Energy Storage Method for Inductor and Capacitor Often time, it is more expedient to think of inductors and capacitors as energy storage devices. This enables us to identify stray (also called parasitic) inductances and capacitances more 4The resisitivity ρ = 1/σ where ρ has the unit of ohm-m, while σ has the unit of siemen/m. Circuit Theory Revisited 265 easily. This manner of thinking allows for an alternative way of calculating inductances and capacitances as well [31]. The energy stored in an inductor is due to its energy storage in the magnetic field, and it is alternatively written, according to circuit theory, as 1 W = LI2 (24.7.1) m 2 Therefore, it is simpler to think that an inductance exists whenever there is stray magnetic field to store magnetic energy. A piece of wire carries a current that produces a magnetic field enabling energy storage in the magnetic field. Hence, a piece of wire in fact behaves like a small inductor, and it is non-negligible at high frequencies: Stray inductances occur whenever there are stray magnetic fields. By the same token, a capacitor can be thought of as an electric energy storage device rather than a charge storage device. The energy stored in a capacitor, from circuit theory, is 1 W = CV 2 (24.7.2) e 2 Therefore, whenever stray electric field exists, one can think of stray capacitances as we have seen in the case of fringing field capacitances in a microstrip line. 24.8 Finding Closed-Form Formulas for Inductance and Capacitance Finding closed form solutions for inductors and capacitors is a difficult endeavor. As in solving Maxwell’s equations or the waveguide proble, only certain geometries are amenable to closed form solutions. Even a simple circular loop does not have a closed form solution for its inductance L. If we assume a uniform current on a circular loop, in theory, the magnetic field can be calculated using Bio-Savart law that we have learnt before, namely that ¢ I(r0)dl0 × Rˆ H(r) = (24.8.1) 4πR2 But the above cannot be evaluated in closed form save in terms of complicate elliptic integrals [117, 133]. Thus it is simpler to just measure the inductance. However, if we have a solenoid as shown in Figure 24.7, an approximate formula for the inductance L can be found if the fringing field at the end of the solenoid can be ignored. The inductance can be found using the flux linkage method [29, 31]. Figure 24.8 shows the schematic used to find the approximate inductance of this inductor. 266 Electromagnetic Field Theory Figure 24.7: The flux-linkage method is used to estimate the inductor of a solenoid (courtesy of SolenoidSupplier.Com). Figure 24.8: Finding the inductor flux linkage approximately by assuming the magnetic field is uniform inside a long solenoid. The capacitance of a parallel plate capacitor can be found by solving a boundary value problem (BVP) for electrostatics as shown in Section 3.3.4. The electrostatic BVP for ca- pacitor involves Poisson’s equation and Laplace equation which are scalar equations [48]. Alternatively, variational expressions can be used to find the lower and upper bounds of capacitors using, for example, Thomson’s theorem [48]. Circuit Theory Revisited 267 Figure 24.9: The capacitance between two charged conductors can be found by solving a boundary value problem (BVP) involving Laplace equation as discussed in 3.3.4. Assume a geometry of two conductors charged to +V and −V volts as shown in Figure 24.9. Surface charges will accumulate on the surfaces of the conductors. Using Poisson’s equations, and Green’s function for Poisson’s equation, one can express the potential in between the two conductors as due to the surface charges density σ(r). It can be expressed as ¢ 0 1 0 σ(r ) Φ(r) = dS 0 (24.8.2) ε S 4π|r − r | where S = S1 + S2 is the union of two surfaces S1 and S2. Since Φ has values of +V and −V on the two conductors, we require that ¢ 0 ( 1 0 σ(r ) +V, r ∈ S1 Φ(r) = dS 0 = (24.8.3) ε S 4π|r − r | −V, r ∈ S2 In the above, σ(r0), the surface charge density, is the unknown yet to be sought and it is embedded in an integral. But the right-hand side of the equation is known. Hence, this equation is also known as an integral equation. The integral equation can be solved by numerical methods. Having found σ(r), then it can be integrated to find Q, the total charge on one of the conductors. Since the voltage difference between the two conductors is known, the capacitance can be found as C = Q/(2V ). 24.9 Importance of Circuit Theory in IC Design The clock rate of computer circuits has peaked at about 3 GHz due to the resistive loss, or the I2R loss. At this frequency, the wavelength is about 10 cm. Since transistors and circuit components are shrinking due to the compounding effect of Moore’s law, most components, which are of nanometer dimensions, are much smaller than the wavelength. Thus, most of the physics of electromagnetic signal in a microchip circuit can be captured using circuit theory. 268 Electromagnetic Field Theory Figure 24.10 shows the schematic and the cross section of a computer chip at different levels: with the transistor level at the bottom-most. The signals are taken out of a transistor by XY lines at the middle level that are linked to the ball-grid array at the top-most level of the chip. And then, the signal leaves the chip via a package. Since these nanometer-size structures are much smaller than the wavelength, they are usually modeled by lumped R, L, and C elements when retardation effect can be ignored. If retardation effect is needed, it is usually modeled by a transmission line. This is important at the package level where the dimensions of the components are larger. A process of parameter extraction where computer software or field solvers (software that solve Maxwell’s equations numerically) are used to extract these lumped-element parameters. Finally, a computer chip is modeled as a network involving a large number of transistors, diodes, and R, L, and C elements. Subsequently, a very useful commercial software called SPICE (Simulation Program with Integrated-Circuit Emphasis) [134], which is a computer- aided software, solves for the voltages and currents in this network. Figure 24.10: Cross section of a chip (top left) and the XY lines in the chip (top right), and the interconnects in the package needed to take the signal out of the chip (bottom right) (courtesy of Wikipedia and Intel). Initially, SPICE software was written primarily to solve circuit problems. But the SPICE software now has many capabilities, including modeling of transmission lines for microwave engineering, which are important for modeling retardation effects. Figure 24.11 shows an interface of an RF-SPICE that allows the modeling of transmission line with a Smith chart interface. Circuit Theory Revisited 269 Figure 24.11: SPICE is also used to solve RF problems. A transmission line is used in com- bination with circuit theory to account for retardation effects in a computer circuit (courtesy of EMAG Technologies Inc.). 24.10 Decoupling Capacitors and Spiral Inductors Decoupling capacitor is an important part of modern computer chip design. They can regulate voltage supply on the power delivery network of the chip as they can remove high-frequency noise and voltage fluctuation from a circuit as shown in Figure 24.12. Figure 24.13 shows a 3D IC computer chip where decoupling capacitors are integrated into its design. Figure 24.12: A decoupling capacitor is essentially a low-pass filter allowing low-frequency signal to pass through, while high-frequency signal is short-circuited (courtesy learningabout- electronics.com). 270 Electromagnetic Field Theory Figure 24.13: Modern computer chip design is 3D and is like a jungle. There are different levels in the chip and they are connected by through silicon vias (TSV). IMD stands for inter- metal dielectrics. One can see different XY lines serving as power and ground lines (courtesy of Semantic Scholars). Inductors are also indispensable in IC design, as they can be used as a high frequency choke. However, designing compact inductor is still a challenge. Spiral inductors are used because of their planar structure and ease of fabrication. However, miniaturizing inductor is a difficult frontier research topic [135]. Figure 24.14: Spiral inductors are difficult to build on a chip, but by using laminal structure, it can be integrated into the IC fabrication process (courtesy of Quan Yuan, Research Gate). Lecture 25 Radiation by a Hertzian Dipole Radiation of electromagnetic field is of ultimate importance for wireless communication sys- tems. The first demonstration of the wave nature of electromagnetic field was by Heinrich Hertz in 1888 [18], some 23 years after Maxwell’s equations were fully established. Guglielmo Marconi, after much perserverence with a series of experiments, successfully transmitted wireless radio signal from Cornwall, England to Newfoundland, Canada in 1901 [136]. The experiment was serendipitous since he did not know that the ionosphere was on his side: The ionosphere helped to bounce the radio wave back to earth from outer space. Marconi’s success ushered in the age of wireless communication, which is omni-present in our daily lives. Hence, radiation by arbitrary sources is an important topic for antennas and wireless communications. We will start with studying the Hertzian dipole which is the simplest of radiation sources we can think of. 25.1 History The original historic Hertzian dipole experiment is shown in Figure 25.1. It was done in 1887 by Heinrich Hertz [18]. The schematic for the original experiment is also shown in Figure 25.2. A metallic sphere has a capacitance in closed form with respect to infinity or a ground plane.1 Hertz could use those knowledge to estimate the capacitance of the sphere, and also, he could estimate the inductance of the leads that are attached to the dipole, and hence, the resonance frequency of his antenna. The large sphere is needed to have a large capacitance, so that current can be driven through the wires. As we shall see, the radiation strength of the dipole is proportional to p = ql the dipole moment. 1We shall learn later that this problem can be solved in closed form using image theorem. 271 272 Electromagnetic Field Theory Figure 25.1: Hertz’s original experiment on a small dipole (courtesy of Wikipedia [18]). Figure 25.2: More on Hertz’s original experiment on a small dipole (courtesy of Wikipedia [18]). The antenna was powered by a transformer. The radiated electromagnetic field was picked up by a loop receiver antenna that generates a spark at its gap M. Radiation by a Hertzian Dipole 273 25.2 Approximation by a Point Source Figure 25.3: Schematic of a small Hertzian dipole which is a close approximation of that first proposed by Hertz. Figure 25.3 is the schematic of a small Hertzian dipole resembling the original dipole that Hertz made. Assuming that the spheres at the ends store charges of value q, and l is the effective length of the dipole, then the dipole moment p = ql. The charge q is varying in time harmonically because it is driven by the generator. Since dq = I, dt we have the current moment dq Il = l = jωql = jωp (25.2.1) dt for this Hertzian dipole. A Hertzian dipole is a dipole which is much smaller than the wavelength under consid- eration so that we can approximate it by a point current distribution, or a current density. Mathematically, it is given by [32, 44] J(r) =zIlδ ˆ (x)δ(y)δ(z) =zIlδ ˆ (r) (25.2.2) The dipole is as shown in Figure 25.3 schematically. As long as we are not too close to the dipole so that it does not look like a point source anymore, the above is a good mathematical model and approximation for describing a Hertzian dipole. 274 Electromagnetic Field Theory Figure 25.4: The field of a point dipole field versus that of a dipole field. When one is far away from the dipole sources, their fields are similar to each other (courtesy of Wikepedia). We have learnt previously that the vector potential is related to the current as follows: ¦ 0 e−jβ|r−r | A(r) = µ dr0J(r0) (25.2.3) 4π|r − r0| Since the current is a 3D delta function in space, using the sifting property of a delta function, the corresponding vector potential is given by µIl A(r) =z ˆ e−jβr (25.2.4) 4πr Since the vector potential A(r) is cylindrically symmetric, the corresponding magnetic field is obtained, using cylindrical coordinates, as 1 1 1 ∂ ∂ H = ∇ × A = ρˆ A − φˆ A (25.2.5) µ µ ρ ∂φ z ∂ρ z ∂ p 2 2 where ∂φ = 0, r = ρ + z . In the above, we have used the chain rule that ∂ ∂r ∂ ρ ∂ ρ ∂ = = = . ∂ρ ∂ρ ∂r pρ2 + z2 ∂r r ∂r As a result, ρ Il 1 1 H = −φˆ − − jβ e−jβr (25.2.6) r 4π r2 r ρ In spherical coordinates, r = sin θ, and (25.2.6) becomes [32] Il H = φˆ (1 + jβr)e−jβr sin θ (25.2.7) 4πr2 Radiation by a Hertzian Dipole 275 The electric field can be derived using Maxwell’s equations. 1 1 1 ∂ 1 ∂ E = ∇ × H = rˆ sin θH − θˆ rH jω jω r sin θ ∂θ φ r ∂r φ Ile−jβr h i = rˆ2 cos θ(1 + jβr) + θˆsin θ(1 + jβr − β2r2) (25.2.8) jω4πr3 Figure 25.5: Spherical coordinates are used to calculate the fields of a Hertzian dipole. 25.2.1 Case I. Near Field, βr 1 Since βr 1, retardation effect within this short distance from the point dipole can be ignored. Also, we let βr → 0, and keeping the largest terms (or leading order terms in math parlance), then from (25.2.8), with Il = jωp p E =∼ (ˆr2 cos θ + θˆsin θ), βr 1 (25.2.9) 4πr3 For the H field, from (25.2.7), with βr 1, then jωp H = φˆ sin θ (25.2.10) 4πr2 or jβrp η H = φˆ sin θ (25.2.11) 0 4πεr3 Thus, it is seen that η0H E, when βr 1 (25.2.12) 276 Electromagnetic Field Theory where p = ql is the dipole moment.2 The above implies that in the near field, the electric field dominates over the magnetic field. r In the above, βr could be made very small by making λ small or by making ω → 0. The above is like the static field of a dipole. Another viewpoint is that in the near field, the field varies rapidly, and space derivatives are much larger than the time derivative.3 For instance, ∂ ∂ ∂x c∂t Alternatively, we can say that the above is equivalent to ∂ ω ∂x c or that 1 ∂2 ∇2 − ≈ ∇2 c2 ∂t2 In other words, static theory prevails over dynamic theory when βr 1. The above approx- imations are consistent with that the retardation effect is negligible over this lengthscale. 25.2.2 Case II. Far Field (Radiation Field), βr 1 This is also known as the far zone. In this case, retardation effect is important. In other words, phase delay cannot be ignored. Il E =∼ θjωµˆ e−jβr sin θ (25.2.13) 4πr and Il H =∼ φjβˆ e−jβr sin θ (25.2.14) 4πr Eθ ωµ p µ Note that = = = η0. Here, E and H are orthogonal to each other and they are Hφ β both orthogonal to the direction of propagation, as in the case of a plane wave. Or in a word, a spherical wave resembles a plane wave in the far field approximation. 25.3 Radiation, Power, and Directive Gain Patterns The time average power flow in the far field is given by 1 1 η βIl 2 hSi = 2 p Here, η0 = µ/ε. We multiply H by η0 so that the quantities we are comparing have the same unit. 3This is in agreement with our observation that electromagnetic fields are great contortionists: They will deform themselves to match the boundary first before satisfying Maxwell’s equations. Since the source point is very small, the fields will deform themselves so as to satisfy the boundary conditions near to the source region. If this region is small compared to wavelength, the fields will vary rapidly over a small lengthscale compared to wavelength. Radiation by a Hertzian Dipole 277 The radiation field pattern of a Hertzian dipole is the plot of |E| as a function of θ at a constant r. Hence, it is proportional to sin θ, and it can be proved that it is a circle. The radiation power pattern is the plot of hSri at a constant r. Figure 25.6: Radiation field pattern of a Hertzian dipole. It can be shown that the pattern is a circle. Figure 25.7: Radiation power pattern of a Hertzian dipole which is also the same as the directive gain pattern. 278 Electromagnetic Field Theory The total power radiated by a Hertzian dipole is thus given by ¢ ¢ ¢ 2π π π 2 2 η0 βIl 3 P = dφ dθr sin θhSri = 2π dθ sin θ (25.3.2) 0 0 0 2 4π Since ¢ ¢ ¢ π −1 1 4 dθ sin3 θ = − (d cos θ)[1 − cos2 θ] = dx(1 − x2) = (25.3.3) 0 1 −1 3 then 4 βIl 2 η (βIl)2 P = πη = 0 (25.3.4) 3 0 4π 12π The directive gain of an antenna, G(θ, φ), is defined as [32] hS i hS i G(θ, φ) = r = r (25.3.5) hS i P av 4πr2 where P hS i = (25.3.6) av 4πr2 is the power density if the power P were uniformly distributed over a sphere of radius r. Notice that hSavi is independent of angle. Hence, the angular dependence of the directive gain G(θ, φ) is coming from hSri. Substituting (25.3.1) and (25.3.4) into the above, we have 2 η0 βIl sin2 θ 2 4πr 3 2 G(θ, φ) = 2 = sin θ (25.3.7) 1 4 βIl 2 4πr2 3 η0π 4π The peak of G(θ, φ) is known as the directivity of an antenna. It is 1.5 in the case of a Hertzian dipole. If an antenna is radiating isotropically, its directivity is 1, which is the lowest possible value, whereas it can be over 100 for some antennas like reflector antennas (see Figure 25.8). A directive gain pattern is a plot of the above function G(θ, φ) and it resembles the radiation power pattern. Radiation by a Hertzian Dipole 279 Figure 25.8: The gain of a reflector antenna can be increased by deflecting the power radiated in the desired direction by the use of a reflector (courtesy of racom.eu). If the total power fed into the antenna instead of the total radiated power is used in the denominator of (25.3.5), the ratio is known as the power gain or just gain and the pattern is the power gain pattern. The total power fed into the antenna is not equal to the total radiated power because there could be some loss in the antenna system like metallic loss. 25.3.1 Radiation Resistance Engineers love to replace complex systems with simpler systems. Simplicity rules again! This will make interface with electronic driving circuits for the antenna easier to derive. A raw Hertzian dipole, when driven by a voltage source, essentially looks like a capacitor due to the preponderance of electric field energy stored in the dipole field. But at the same time, the dipole radiates giving rise to radiation loss. Thus a simple circuit equivalence of a Hertzian dipole is a capacitor in series with a resistor. The resistor accounts for radiation loss of the dipole. 280 Electromagnetic Field Theory Figure 25.9: (a) Equivalent circuit of a raw Hertzian dipole without matching. (b) Equivalent circuit of a matched Hertzian dipole (using maximum power transfer theorem). (c) Equivalent circuit of a matched dipole at the resonance frequency of the LC tank circuit. Hence, the way to drive the Hertzian dipole effectively is to use matching network for maximum power transfer. Or an inductor has to be added in series with the intrinsic ca- pacitance of the Hertzian dipole to cancel it at the resonance frequency of the tank circuit. Eventually, after matching, the Hertzian dipole can be modeled as just a resistor. Then the 1 2 power absorbed by the Hertzian dipole from the driving source is P = 2 I Rr. Thus, the radiation resistance Rr is the effective resistance that will dissipate the same power as the radiation power P when a current I flows through the resistor. Hence, it is defined by [32] 2P (βl)2 R = = η ≈ 20(βl)2, where η = 377 ≈ 120π Ω (25.3.8) r I2 0 6π 0 For example, for a Hertzian dipole with l = 0.1λ, Rr ≈ 8Ω. The above assumes that the current is uniformly distributed over the length of the Hertzian dipole. This is true if there are two charge reservoirs at its two ends. For a small dipole with no charge reservoir at the two ends, the currents have to vanish at the tips of the dipole as shown in Figure 25.10. The effective length of an equivalent Hertzian dipole for the dipole with triangular distribution is half of its actual length due to the manner the currents are distributed.4 Such a formula can be used to estimate the radiation resistance of a dipole. For example, a half-wave dipole does not have a triangular current distribution a sinusoidal one as shown in Figure 25.11. Nevertheless, we approximate the current distribution of a half- λ wave dipole with a triangular distribution, and apply the above formula. We pick a = 2 , and λ let leff = 4 in (25.3.8), we have Rr ≈ 50Ω (25.3.9) 4As shall be shown, when the dipole is short, the details of the current distribution is inessential in determining the radiation field. It is the area under the current distribution that is important. Radiation by a Hertzian Dipole 281 Figure 25.10: The current pattern on a short dipole can be approximated by a triangle since the current has to vanish at the end points of the short dipole. Furthermore, this dipole can be approximated by an effective Hertzian dipole half its length with uniform current. The true current distribution on a half-wave dipole resembles that shown in Figure 25.11. The current is zero at the end points, but the current has a more sinusoidal-like distribution as in a transmission line. Hence, a half-wave dipole is not much smaller than a wavelength and does not qualify to be a Hertzian dipole. Furthermore, the current distribution on the half-wave dipole is not triangular in shape as above. A more precise calculation shows that Rr = 73Ω for a half-wave dipole [54]. This also implies that a half-wave dipole with sinusoidal current distribution is a better radiator than a dipole with triangular current distribution. In fact, one can think of a half-wave dipole as a flared, open transmission line. In the beginning, this flared open transmission line came in the form of biconical antennas which are shown in Figure 25.12 [137]. If we recall that the characteristic impedance of a transmission line is pL/C, then as the spacing of the two metal pieces becomes bigger, the equivalent characteristic impedance gets bigger. Therefore, the impedance can gradually transform from a small impedance like 50 Ω to that of free space, which is 377 Ω. This impedance matching helps mitigate reflection from the ends of the flared transmission line, and enhances radiation. Because of the matching nature of bicone antennas, they are better radiators with higher radiaton loss and lower Q. Thus they have a broader bandwidth, and are important in UWB (ultra-wide band) antennas [138]. 282 Electromagnetic Field Theory Figure 25.11: Approximate current distribution on a half-wave dipole (courtesy of electronics- notes.co). The currents are zero at the two end tips due to the current continuity equation, or KCL. Figure 25.12: A bicone antenna can be thought of as a transmission line with gradually changing characteristic impedance. This enhances impedance matching and the radiation of the antenna (courtesy of antennasproduct.com). Lecture 26 Radiation Fields Figure 26.1: (a) Electric field around a time-oscillating dipole (courtesy of physics stack exchange). (b) Equi-potential lines aroud a moving charge that gives rise Cherenkov radiation (courtesy of J.D. Jackson [48]). We will not study Cherenkov (Cerenkov) radiation in this course, but it is written up in [48] and [32]. Its was a Nobel Prize winning discovery. The reason why charges radiate is because they move or accelerate. In the case of a dipole antenna, the charges move back and forth between poles of the antenna. Near to the dipole source, quasi-static physics prevails, and the field resembles that of a static dipole. If the dipole is flipping sign constantly due to the change in the direction of the current flow, the field would also have to flip sign constantly. But electromagnetic wave travels with a finite velocity. The field from the source ultimately cannot keep up with the sign change of the source field: it has to be ‘torn’ away from the source field and radiate. Another interesting radiation is the Cherenkov radiation. It is due to a charge moving faster than the velocity of light. As an electron cannot move faster than the speed of light in a vacuum, this can only 283 284 Electromagnetic Field Theory happen in the material media or plasma, where the velocity of the electron can be faster than the group velocity of wave in the medium. Ultimately, the electric field from the particle is ‘torn’ off from the charge and radiate. These two kinds of radiation are shown in the Figure 26.1. We have shown how to connect the vector and scalar potentials to the sources J and % of an electromagnetic system. This is a very important connection: it implies that once we know the sources, we know how to find the fields. But the relation between the fields and the sources are in general rather complex. In this lecture, we will simplify this relation by making a radiation field or far-field approximation by assuming that the point where the field is observed is very far from the source location in terms of wavelegth. This approximation is very useful for understanding the physics of the radiation field from a source such as an antenna. It is also important for understanding the far field of an optical system. As shall be shown, this radiation field carries the energy generated by the sources to infinity. 26.1 Radiation Fields or Far-Field Approximation Figure 26.2: The relation of the observation point located a r to the source location at r0. The distance of the observation point r to the source location r0 is |r − r0|. In the previous lecture, we have derived the relation of the vector and scalar potentials to the sources J and % as shown in (23.2.29) and (23.2.30)1 They are given by ¦ 0 e−jβ|r−r | A(r) = µ dr0J(r0) (26.1.1) 4π|r − r0| ¦V −jβ|r−r0| 1 0 0 e Φ(r) = dr %(r ) 0 (26.1.2) ε V 4π|r − r | 1This topic is found in many standard textbooks in electromagnetics [32, 47, 54]. They are also found in lecture notes [44, 139]. Radiation Fields 285 √ where β = ω µε = ω/c is the wavenumber. The integrals in (26.1.1) and (26.1.2) are normally untenable, but when the observation point is far from the source, approximation to the integrals can be made giving them a nice physical interpretation. Figure 26.3: The relation between |r| and |r−r0| using the parallax method, or that |r−r0| ≈ |r| − r0 · rˆ. It is assumed that r is almost parallel to r − r0. 26.1.1 Far-Field Approximation When |r| |r0|, then |r − r0| ≈ r − r0 · rˆ, where r = |r| . This approximation can be shown algebraically or by geometrical argument as shown in Figure 26.3. Thus (26.1.1) above becomes ¦ ¦ 0 −jβr µ 0 J(r ) −jβr+jβr0·rˆ µe 0 0 jβr0·rˆ A(r) ≈ dr 0 e ≈ dr J(r )e (26.1.3) 4π V r − r · rˆ 4πr V In the above, r0 · rˆ is small compared to r. Hence, we have made use of that 1/(1 − ∆) ≈ 1 when ∆ is small, so that 1/(r − r0 · rˆ) can be approximate by 1/r. Also, we assume that the 0 frequency is sufficiently high such that βr0 ·rˆ is not necessarily small. Thus, ejβr ·rˆ 6= 1, unless βr0 · rˆ 1. Hence, we keep the exponential term in (26.1.3) but simplify the denominator to arrive at the last expression above. If we let β = βrˆ, which is the β vector (or k vector in optics), and r0 =xx ˆ 0 +yy ˆ 0 +zz ˆ 0, then 0 0 0 0 0 ejβr ·rˆ = ejββ·r = ejβxx +jβy y +jβz z (26.1.4) Therefore (26.1.3) resembles a 3D Fourier transform integral,2 namely, the above integral 2Except that the vector β is of fixed length. 286 Electromagnetic Field Theory becomes ¦ −jβr µe 0 A(r) ≈ dr0J(r0)ejβ·r (26.1.5) 4πr V and (26.1.5) can be rewritten as µe−jβr A(r) =∼ F(β) (26.1.6) 4πr where ¦ 0 F(β) = dr0J(r0)ejβ·r (26.1.7) V is the 3D Fourier transform of J(r0) with the Fourier transform variable β =rβ ˆ . 2 2 2 It is to be noted that this is not a normal 3D Fourier transform because |β| = βx +βy + 2 2 βz = β which is a constant for a fixed frequency. In other words, the length of the vector β is fixed to be β, whereas in a normal 3D Fourier transform, βx, βy, and βz are independent 2 2 2 variables, each with values in the range [−∞, ∞]. Or the value of βx +βy +βz ranges from zero to infinity. The above is the 3D “Fourier transform” of the current source J(r0) with Fourier variables, βx, βy, βz restricted to lying on a sphere of radius β and β = βrˆ. This spherical surface in the Fourier space is also called the Ewald sphere. 26.1.2 Locally Plane Wave Approximation We can writer ˆ or β in terms of direction cosines in spherical coordinates or that rˆ =x ˆ cos φ sin θ +y ˆsin φ sin θ +z ˆcos θ (26.1.8) Hence, F(β) = F(βrˆ) = F(β, θ, φ) (26.1.9) It is not truly a 3D function, since β, the length of the vector β, is fixed. It is a 3D Fourier transform with data restricted on a spherical surface. Also in (26.1.6), when r r0 · rˆ, and when the frequency is high or β is large, e−jβr is now a rapidly varying function of r while, F(β) is only a slowly varying function ofr ˆ or of θ and φ, the observation angles. In other words, the prefactor in (26.1.6), exp(−jβr)/r, can be thought of as resembling a spherical wave. Hence, if one follows a ray of this spherical wave and moves in the r direction, the predominant variation of the field is due to e−jβr, whereas the direction of the vector β changes little, and hence, F(β) changes little. Furthermore, r0 in (26.1.7) are restricted to small or finite number, making F(β) a weak function of β (see Figure 26.4). The above shows that in the far field, the wave radiated by a finite source resembles a spherical wave. Moreover, a spherical wave resembles a plane wave when one is sufficiently far from the source such that βr 1, or 2πr/λ 1. Or r is many wavelengths away from Radiation Fields 287 the source. Hence, we can write e−jβr = e−jββ·r where β =rβ ˆ and r =rr ˆ so that a spherical wave resembles a plane wave locally. This phenomenon is shown in Figure 26.4 and Figure 26.5 Figure 26.4: A source radiates a field that resembles a spherical wave. In the vicinity of the observation point r, when β is large, the field is strongly dependent on r via exp(−jβr) but weakly dependent on β (beta hardly changes direction in the vicinity of the observation point). Hence, the field becomes locally a plane wave in the far field. 288 Electromagnetic Field Theory Figure 26.5: (a) A leaky hole in a waveguide leaks a spherical (courtesy of MEEP, MIT). (b) A point source radiates a spherical wave (courtesy of ME513, Purdue Engineering). Most of these simulations are done with FDTD (finite-difference time-domain) method that we will learn later in the course. When the wavelength is short, or the frequency high, a spherical wave front looks locally plane. This is similar to the notion that as humans, who are short, think that the earth is flat around us. Up to this day, some people still believe that the earth is flat:) Then, it is clear that with the local plane-wave approximation, ∇ → −jββ = −jβrˆ, we have 1 β β H = ∇ × A ≈ −j rˆ × (θAˆ + φAˆ ) = j (θAˆ − φAˆ ) (26.1.10) µ µ θ φ µ φ θ Similarly [44, 139], 1 β E = ∇ × H =∼ −j rˆ × H =∼ −jω(θAˆ + φAˆ ) (26.1.11) jωε ω θ φ Notice that β = βrˆ, the direction of propagation of the local plane wave, is orthogonal to E and H in the far field, a property of a plane wave since the wave is locally a plane wave. Moreover, there are more than one way to derive the electric field E. For instance, using (26.1.10) for the magnetic field, the electric field can also be written as 1 E = ∇ × ∇ × A (26.1.12) jωµε Using the formula for the double-curl operator, the above can be rewritten as 1 1 E = ∇∇ · A − ∇2A =∼ −ββ + β2I · A (26.1.13) jωµε jωµε Radiation Fields 289 where we have used that ∇ =∼ −jβ and ∇2A = −β2A.3 Alternatively, we can factor β2 = ω2µ out of the parenthesis, and rewrite the above as E =∼ −jω −βˆβˆ + I · A = −jω −rˆrˆ + I · A (26.1.14) Since I =r ˆrˆ + θˆθˆ + φˆφˆ, then the above becomes ∼ ˆˆ ˆˆ ˆ ˆ E = −jω θθ + φφ · A = −jω(θAθ + φAφ) (26.1.15) which is the same as previously derived. It also shows that the electric field is transverse to the β vector.4 Furthermore, it can be shown that in the far field, using the local plane-wave approxima- tion, |E|/|H| ≈ η (26.1.16) where η is the intrinsic impedance of free space, which is a property of a plane wave. Moreover, one can show that the time average Poynting’s vector, or the power density flow, in the far field is 1 1 hSi = 26.1.3 Directive Gain Pattern Revisited We have defined the directive gain pattern for a Hertzian dipole before in Section 25.3. But this concept can be applied to a general radiating source or antenna. Once the far-field radiation power pattern or the radial power density hSri is known, the total power radiated by the antenna in the far field can be found by integrating over all angles, viz., ¢ ¢ π 2π 2 PT = r sin θdθdφhSr(θ, φ)i (26.1.19) 0 0 3Note that ∇ · A 6= 0 here. 4We can also arrive at the above by letting E = −jωA − ∇Φ, and using the appropriate formula for the scalar potential. There is more than one road that lead to Rome! 5To avoid confusion, we will use S to denote instantaneous Poynting’s vector and S to denote complex Poynting’s vector (see 10.3.1). e 290 Electromagnetic Field Theory The above evaluates to a constant independent of r due to energy conservation. Now assume that this same antenna is radiating isotropically in all directions, then the average power density of this fictitious isotropic radiator as r → ∞ is P hS i = T (26.1.20) av 4πr2 A dimensionless directive gain pattern can be defined as before in Section 25.3 such that [32, 139] hS (θ, φ)i 4πr2hS (θ, φ)i G(θ, φ) = r = r (26.1.21) hSavi PT This directive gain pattern is a measure of the radiation power pattern of the antenna or source compared to when it radiates isotropically. The above function is independent of r in 2 the far field since Sr ∼ 1/r in the far field. As in the Hertzian dipole case, the directivity of an antenna D = max(G(θ, φ)), is the maximum value of the directive gain. It is to be noted that by its mere definition, ¢ dΩG(θ, φ) = 4π (26.1.22) ¡ ¡ ¡ 2π π where dΩ = 0 0 sin θdθdφ. It is seen that since the directive gain pattern is normalized, when the radiation power is directed to the main lobe of the antenna, the corresponding side lobes and back lobes will be diminished. An antenna also has an effective area or aperture Ae, such that if a plane wave carrying power density denoted by hSinci impinges on the antenna, then the power received by the antenna, Preceived is given by Preceived = hSinciAe (26.1.23) Here, the transmit antenna and the receive antenna are in the far field of each other. Hence, we can approximate the field from the transmit antenna to be a plane wave when it reaches the receive antenna. If the receive antenna is made of PEC, induced current will form on the receive antenna so as to generated a field that will cancel the incident field on the PEC surface. This induced current generates a voltage at the receiver load, and hence power received by the antenna. A wonderful relationship exists between the directive gain pattern G(θ, φ) and the effective aperture, namely that6 λ2 A = G(θ, φ) (26.1.24) e 4π Therefore, the effective aperture of an antenna is also direction dependent. The above im- plies that the radiation property of an antenna is related to its receiving property. This is a 6The proof of this formula is beyond the scope of this lecture, but we will elaborate on it when we discuss reciprocity theorem. Radiation Fields 291 beautiful consequence of reciprocity theorem that we will study later! The constant of pro- portionality, λ2/(4π) is a universal constant that is valid for all antennas satisfying reciprocity theorem. The derivation of this constant for a Hertzian dipole is given in Kong [32], or using blackbody radiation law [139,140]. The directivity and the effective aperture can be enhanced by designing antennas with different gain patterns. When the radiative power of the antenna can be directed to be in a certain direction, then the directive gain and the effective aperture (for that given direction) of the antenna is improved. This is shown in Figure 26.6. Such focussing of the radiation fields of the antenna can be achieved using reflector antennas or array antennas. Array antennas, as shall be shown, work by constructive and destructive wave field of the antenna. Being able to do point-to-point communications at high data rate is an important modern application of antenna array. Figure 26.7 shows the gain pattern of a sophisticated antenna array design for 5G applications. Figure 26.6: The directive gain pattern of an array antenna. The directivity is increased by constructive interference (courtesy of Wikepedia). 292 Electromagnetic Field Theory Figure 26.7: The directive gain pattern of a sophisticated array antenna for 5G applications (courtesy of Ozeninc.com). Lecture 27 Array Antennas, Fresnel Zone, Rayleigh Distance Our world is beset with the dizzying impact of wireless communication. It has greatly im- pact our lives.1 Wireless communication is impossible without using antennas. Hence it is important to design these communication systems with utmost efficiency and sensitivity. We have seen that a simple Hertzian dipole has low directivity in Section 25.3. The radiation pattern looks like that of a donut, and the directivity of the antenna is 1.5. Hence, for point- to-point communications, much power is wasted. However, the directivity of antennas can be improved if a group or array of dipoles can work cooperatively together. They can be made to constructively interfere in the desired direction, and destructively interfere in other directions to enhance their directivity. Since the far-field approximation of the radiation field can be made, and the relationship between the far field and the source is a Fourier transform relationship, clever engineering can be done borrowing knowledge from the signal processing area. After understanding the far-field physics, one can also understand many optical phe- nomena, such as how a laser pointer works. Many textbooks have been written about array antennas some of which are [141, 142]. 27.1 Linear Array of Dipole Antennas Antenna array can be designed so that the constructive and destructive interference in the far field can be used to steer the direction of radiation of the antenna, or the far-field radiation pattern of an antenna array. This is because the far field of a source is related to the source by a Fourier transform relationship. The relative phases of the array elements can be changed in time so that the beam of an array antenna can be steered in real time. This has important applications in, for example, air-traffic control. It is to be noted that if the current sources are impressed current sources, and they are the input to Maxwell’s equations, and the fields are 1I cannot imagine a day that I do not receive messages from my relatives and friends in Malaysia and Singapore. It has made the world look smaller, and more transparent. 293 294 Electromagnetic Field Theory the output of the system, then we are dealing with a linear system here whereby linear system theory can be used. For instance, we can use Fourier transform to analyze the problem in the frequency domain. The time domain response can be obtained by inverse Fourier transform. This is provided that the current sources are impressed and they are not affected by the fields that they radiate. Figure 27.1: Schematics of a dipole array. To simplify the math, the far-field approximation can be used to find its far field or radiation field. A simple linear dipole array is shown in Figure 27.1. First, without loss of generality and for simplicity to elucidate the physics, we assume that this is a linear array of point Hertzian dipoles aligned on the x axis. The current can then be described mathematically as follows: 0 0 0 0 J(r ) =zIl ˆ [A0δ(x ) + A1δ(x − d1) + A2δ(x − d2) + ··· 0 0 0 + AN−1δ(x − dN−1)]δ(y )δ(z ) (27.1.1) The far field can be found using the approximate formula derived in the previous lecture, viz., (26.1.3) reproduced below: ¦ −jβr µe 0 A(r) ≈ dr0J(r0)ejβ·r (27.1.2) 4πr V To reiterate, the above implies that the far field is related to the Fourier transform of the current source J(r0). 27.1.1 Far-Field Approximation The vector potential on the xy-plane in the far field, using the sifting property of delta function, yield the following equation for A(r) using (27.1.2), ¦ µIl 0 A(r) =∼ zˆ e−jβr dr0[A δ(x0) + A δ(x0 − d ) + ··· ]δ(y0)δ(z0)ejβr ·rˆ (27.1.3) 4πr 0 1 1 In the above, for simplicity, we will assume that the observation point is on the xy plane, or that r = ρ =xx ˆ +yy ˆ . Thus,r ˆ =x ˆ cos φ +y ˆsin φ. Also, since the sources are aligned on the Array Antennas, Fresnel Zone, Rayleigh Distance 295 0 0 x axis, then r0 =xx ˆ 0, and r0 · rˆ = x0 cos φ. Consequently, ejβr ·rˆ = ejβx cos φ. By so doing, we have µIl A(r) =∼ zˆ e−jβr[A + A ejβd1 cos φ + A ejβd2 cos φ + ··· + A ejβdN−1 cos φ] (27.1.4) 4πr 0 1 2 N−1 Next, for further simplification, we let dn = nd, implying an equally space array with dis- jnψ tance d between adjacent elements. Then we let An = e , which assumes a progressively increasing phase shift between different elements. Such an antenna array is called a linear phase array. Thus, (27.1.3) in the above becomes µIl A(r) =∼ zˆ e−jβr[1 + ej(βd cos φ+ψ) + ej2(βd cos φ+ψ) + ··· 4πr +ej(N−1)(βd cos φ+ψ)] (27.1.5) With the simplifying assumptions, the above series can be summed in closed form. 27.1.2 Radiation Pattern of an Array The above(27.1.5) can be summed in closed form using the formula N−1 X 1 − xN xn = (27.1.6) 1 − x n=0 Then in the far field, µIl 1 − ejN(βd cos φ+ψ) A(r) =∼ zˆ e−jβr (27.1.7) 4πr 1 − ej(βd cos φ+ψ) ˆ ˆ Ordinarily, as shown previously in (26.1.11), E ≈ −jω(θAθ +φAφ). But since A isz ˆ directed, Aφ = 0. Furthermore, on the xy plane, Eθ ≈ −jωAθ = jωAz. As a consequence, jN(βd cos φ+ψ) 1 − e |Eθ| = |E0| , r → ∞ 1 − ej(βd cos φ+ψ) sin N (βd cos φ + ψ) = |E | 2 , r → ∞ (27.1.8) 0 1 sin 2 (βd cos φ + ψ) The factor multiplying |E0| above is also called the array factor. The above can be used to plot the far-field pattern of an antenna array. Equation (27.1.8) has an array factor that is of the form |sin (Nx)| |sin x| This function appears in digital signal processing frequently, and is known as the digital sinc function [143]. The reason why this is so is because the far field is proportional to the Fourier transform of the current. The current in this case a finite array of Hertzian dipole, which is 296 Electromagnetic Field Theory a product of a box function and infinite array of Hertzian dipole. The Fourier transform of such a current, as is well known in digital signal processing, is the digital sinc. |sin(3x)| Plots of |sin(3x)| and |sin x| are shown as an example and the resulting |sin x| is also shown in Figure 27.2. The function peaks when both the numerator and the denominator of the digital sinc vanish. The value can be found by Taylor series expansion. This happens when x = nπ for integer n. Such an apparent singularity is called a removable singularity. |sin 3x| Figure 27.2: Plot of the digital sinc, |sin x| . 1 In equation (27.1.8), x = 2 (βd cos φ+ψ). We notice that the maximum in (27.1.8) would occur if x = nπ, or if βd cos φ + ψ = 2nπ, n = 0, ±1, ±2, ±3, ··· (27.1.9) The zeros or nulls will occur at Nx = nπ, or 2nπ βd cos φ + ψ = , n = ±1, ±2, ±3, ··· , n 6= mN (27.1.10) N For example, π Case I. ψ = 0, βd = π, principal maximum is at φ = ± 2 . If N = 5, nulls are at −1 2n ◦ ◦ ◦ ◦ φ = ± cos 5 , or φ = ±66.4 , ±36.9 , ±113.6 , ±143.1 . The radiation pattern is seen to form lobes. The largest lobe is called the main lobe, while the smaller lobes are called side lobes. Since ψ = 0, the radiated fields in the y direction are in phase and the peak of the radiation lobe is in the y direction or the broadside direction. Hence, this is called a broadside array. The radiation pattern of such an array is shown in Figure 27.3. Case II. ψ = π, βd = π, principal maximum is at φ = 0, π. If N = 4, nulls are at −1 n ◦ ◦ ◦ ◦ φ = ± cos 2 − 1 , or φ = ±120 , ±90 , ±60 . Since the sources are out of phase by 180 , and N = 4 is even, the radiation fields cancel each other in the broadside, but add in the x Array Antennas, Fresnel Zone, Rayleigh Distance 297 direction or the end-fire direction. This is called the endfire array. Figure 27.4 shows the radiation pattern of such an array. Figure 27.3: The radiation pattern of a five-element broadside array. The broadside and endfire directions of the array are also labeled. Figure 27.4: By changing the phase of the linear array, the radiation pattern of the antenna array can be changed to become an endfire array. From the above examples, it is seen that the interference effects between the different antenna elements of a linear array focus the power in a given direction. We can use linear array to increase the directivity of antennas. Moreover, it is shown that the radiation patterns can be changed by adjusting the spacings of the elements as well as the phase shift between 298 Electromagnetic Field Theory them. The idea of antenna array design is to make the main lobe of the pattern to be much higher than the side lobes so that the radiated power of the antenna is directed along the main lobe or lobes rather than the side lobes. So side-lobe level suppression is an important goal of designing a highly directive antenna. Also, by changing the phase of the antenna elements in real time, the beam of the antenna can be steered in real time with no moving parts. 27.2 When is Far-Field Approximation Valid? In making the far-field approximation in (27.1.3), it will be interesting to ponder when the far-field approximation is valid? That is, when we can approximate 0 0 e−jβ|r−r | ≈ e−jβr+jβr ·rˆ (27.2.1) to arrive at (27.1.3). This is especially important because when we integrate over r0, it can range over large values especially for a large array. In this case, r0 can be as large as (N −1)d. The above approximation is important also because it tells when the field generated by an array antenna become a spherical wave. To answer this question, we need to study the approximation in (27.2.1) more carefully. First, we have |r − r0|2 = (r − r0) · (r − r0) = r2 − 2r · r0 + r02 (27.2.2) We can take the square root of the above to get !1/2 2r · r0 r02 |r − r0| = r 1 − + (27.2.3) r2 r2 Next, we use the Taylor series expansion to get, for small x, that n(n − 1) (1 + x)n ≈ 1 + nx + x2 + ··· (27.2.4) 2! or that 1 1 (1 + x)1/2 ≈ 1 + x − x2 + ··· (27.2.5) 2 8 We can apply this approximation by letting 2r · r0 r02 x =∼ − + r2 r2 To this end, we arrive at2 " # r · r0 1 r02 1 r · r0 2 |r − r0| ≈ r 1 − + − + ··· (27.2.6) r2 2 r2 2 r2 2The art of making such approximation is called perturbation expansion [45]. Array Antennas, Fresnel Zone, Rayleigh Distance 299 In the above, we have not kept every term of the x2 terms by assuming that r02 r0 · r, and terms much smaller than the last term in (27.2.6) can be neglected. We can multiply out the right-hand side of the above to further arrive at r · r0 1 r02 1 (r · r0)2 |r − r0| ≈ r − + − + ··· r 2 r 2 r3 1 r02 1 = r − rˆ · r0 + − (ˆr · r0)2 + ··· (27.2.7) 2 r 2r The last two terms in the last line of (27.2.7) are of the same order.3 Moreover, their sum is bounded by r02/(2r) sincer ˆ · r0 is always less than r0. Hence, the far field approximation is valid if r02 β 1 (27.2.8) 2r In the above, β is involved because the approximation has to be valid in the exponent, namely exp(−jβ|r − r0|) where β multiplies |r − r0| or its approximation. If (27.2.8) is valid, then 02 jβ r e 2r ≈ 1 and thus, the first two terms on the right-hand side of (27.2.7) suffice to approximate |r − r0| on the left-hand side, which are the two terms we have kept in the far-field approximation. 27.2.1 Rayleigh Distance Figure 27.5: The right half of a Gaussian beam [82] displays the physics of the near field, the Fresnel zone, and the far zone. In the far zone, the field behaves like a spherical wave. 3The math parlance for saying that these two terms are approximately of the same magnitude as each other. 300 Electromagnetic Field Theory If we have an infinite time-harmonic current sheet, it can be shown that by matching boundary conditions, it will launch plane waves on both sides of the current sheet [32][p. 652]. Thus if we have an aperture antenna like the opening of a waveguide that is much larger than the wavelength, it will launch a wave that is almost like a plane wave from it. Thus, when a wave field leaves an aperture antenna, it can be approximately described by a Gaussian beam [82] (see Figure 27.5). Near to the antenna aperture, or in the near zone, it is approximately a plane wave with wave fronts parallel to the aperture surface. Far from the antenna aperture, or in the far zone, the field behaves like a spherical wave, with its typical wave front. In between, we are in the Fresnel zone. Consequently, after using that β = 2π/λ, for the far-field approximation to be valid, we need (27.2.8) to be valid, or that π r r02 (27.2.9) λ 0 ∼ If the aperture of the antenna is of radius W , then r < rmax = W and the far field approxi- mation is valid if π r W 2 = r (27.2.10) λ R If r is larger than this distance, then an antenna beam behaves like a spherical wave and starts to diverge. This distance rR is also known as the Rayleigh distance. After this distance, the wave from a finite size source resembles a spherical wave which is diverging in all directions (see Figure 27.5). Also, notice that the shorter the wavelength λ, the larger is this distance. This also explains why a laser pointer works. A laser pointer light can be thought of radiation from a finite size source located at the aperture of the laser pointer as shall be shown using equivalence theorem later. The laser pointer beam remains collimated for quite a distance, before it becomes a divergent beam or a beam with a spherical wave front. In some textbooks [32], it is common to define acceptable phase error to be π/8. The Rayleigh distance is the distance beyond which the phase error is below this value. When the phase error of π/8 is put on the right-hand side of (27.2.8), one gets r02 π β ≈ (27.2.11) 2r 8 Using the approximation, the Rayleigh distance is defined to be 2D2 r = (27.2.12) R λ where D = 2W is the diameter of the antenna aperture. This concept is important in both optics and microwave. 27.2.2 Near Zone, Fresnel Zone, and Far Zone Therefore, when a source radiates, the radiation field is divided into the near zone, the Fresnel zone, and the far zone (also known as the radiation zone, or the Fraunhofer zone in optics). Array Antennas, Fresnel Zone, Rayleigh Distance 301 The Rayleigh distance is the demarcation boundary between the Fresnel zone and the far zone. The larger the aperture of an antenna array is, the further one has to be in order to reach the far zone of an antenna. This distance becomes larger too when the wavelength is short. In the far zone, the far field behaves like a spherical wave, and its radiation pattern is proportional to the Fourier transform of the current. In some sources, like the Hertzian dipole, in the near zone, much reactive energy is stored in the electric field or the magnetic field near to the source. This near zone receives reactive power from the source, which corresponds to instantaneous power that flows from the source, but is return to the source after one time harmonic cycle. Thus, reactive power corresponds to energy that sloshes back and forth between the source and the near field. Hence, a Hertzian dipole has input impedance that looks like that of a capacitor, because much of the near field of this dipole is the electric field. The field in the far zone carries power that radiates to infinity. As a result, the field in the near zone decays rapidly, but the field in the far zone decays as 1/r for energy conservation. In other words, the far field convects energy to infinity. 302 Electromagnetic Field Theory Lecture 28 Different Types of Antennas—Heuristics We have studied different closed form solutions and approximate solutions to Maxwell’s equa- tions. Examples of closed form solutions are transmission lines, waveguides, resonators, and dipoles solutions. Examples of approximate solutions are circuit theory and far field approx- imations. These solutions offer us insights into the physical behaviour of electromagnetic fields, and also the physical mechanisms as to how things work. These physical insights often inspire us for new designs. Fortunately for us, Maxwell’s equations are accurate from sub-atomic lengthscales to galactic lengthscales. In vacuum, they have been validated to extremely high accuracy (see Section 1.1). Furthermore, in the last few decades since the 1960s, very many numerical solutions have been possible for Maxwell’s equations of complex structures. This field of solving Maxwell’s equations numerically is known as computational electromagnetics. It shall be discussed later in this course, and many commercial software are available to solve Maxwell’s equations to high fidelity. Therefore, design engineers these days do not require higher knowledge of math and physics, and the solutions of Maxwell’s equations can be obtained by learning how to use these commercial software. This is a boon to many design engineers: by running these software with cut-and-try engineering, wonderful systems can be designed. It used to be said that if we lock 100 monkeys in a room and let them punch at the 100 keyboards, they will never type out Macbeth nor Hamlet. But with 100 engineers trained with good physical insight, when locked up in a room with commercial software, with enough time and patience, they can come up with wonderful designs of different electromagnetic systems. In the parlance of the field, it is known as virtual proto-typing. It is mainly driven by heuristics and cut-and-try engineering. Therefore, we will discuss the functions of different antennas heuristically in this lecture. 303 304 Electromagnetic Field Theory 28.1 Resonance Tunneling in Antenna We realize the power of resonance enhancement when we were young by playing on a swing in the park. By pumping the swing at its resonance frequency, we can cause it to swing at a large amplitude without a Herculean effort. A simple antenna like a short dipole behaves like a Hertzian dipole with an effective length. A short dipole has an input impedance resembling that of a capacitor. Hence, it is difficult to drive current into the antenna unless other elements are added. Hertz was clever by using two metallic spheres to increase the current flow. A large current flow on the stem of the antenna makes the stem resemble an inductor. Thus, the end-cap capacitances and the stem inductance together can act like a resonator enhancing the current flow on the antenna. Some antennas are deliberately built to resonate with its structure to enhance its radiation. A half-wave dipole is such an antenna as shown in Figure 28.1 [137]. These antennas are using resonance tunneling to increase the currents on them to enhance their radiation efficiencies. A half-wave dipole can also be thought of as a flared open transmission line in order to make it radiate. It can be gradually morphed from a quarter-wavelength transmission line as shown in Figure 28.1. A transmission line is a poor radiator, because the electromagnetic energy is trapped between two pieces of metal. But a flared transmission line can radiate its field to free space. The dipole antenna, though a simple device, has been extensively studied by King [144].1 Figure 28.1: A half-wave dipole can be thought of as a resonator with radiation loss. It can be thought of as a quarter-wavelength transmission line that is gradually opened up (courtesy of electronics-notes.com). 1He has reputed to have produced over 100 PhD students studying the dipole antenna. Different Types of Antennas—Heuristics 305 Figure 28.2: The electromagnetic field around a Goubau line (courtesy of [145]). The field resembles that of a coaxial cable with the outer conductor located at infinity. One can also think of a piece of a wire as a waveguide. It is called a Goubau line as shown in Figure 28.2, which can be thought of the limiting case of a coaxial cable where the outer conductor is infinitely far away [145]. The wave is weakly guided since it now can shed energy to infinity. The behavior of a wire as a Goubau line waveguide can be used to explain heuristically why a half-wave dipole resonates when it is about half wavelength. A folded dipole is often used to alter the input impedance of a dipole antenna [146]. Even though it can have a resonant frequency lower than that of a normal dipole, the lowest resonant mode does not radiate well. The mode that radiates well has the same resonant length as an unfolded dipole. It has a radiation resistance four times that of a half-wave dipole of similar length which is about 300 ohms. This is equal to the characteristic impedance of a twin-lead transmission line [147]. Figure 28.3 shows a Yagi-Uda antenna driven by a folded dipole. 306 Electromagnetic Field Theory Figure 28.3: A Yagi-Uda antenna was invented by heuristics in 1926. The principal element of the antenna is the folded dipole. When a wire dipole antenna is less than half a wavelength, it acts as a waveguide, or a director. When the wire antenna is slightly more than half a wavelength, it acts as a reflector [148]. Therefore, the antenna radiates predominantly in one direction (courtesy of Wikipedia [149]). A Yagi-Uda antenna is also another interesting invention. It was invented in 1926 by Yagi and Uda in Japan by plainly using physical intuition [148]. Physical intuition was a tool of engineers of yesteryears while modern engineers tend to use sophisticated computer-aided design (CAD) software. The principal driver element of the antenna is the folded dipole. Surprisingly, the elements of dipoles in front of the driver element are acting like a waveguide in space, while the element at the back acts like a reflector. Therefore, the field radiated by the driver element will be directed toward the front of the antenna. Thus, this antenna has higher directivity than just a stand alone dipole. Due to its simplicity, this antenna has been made into nano-antennas which operate at optical frequencies [150]. Figure 28.4: A cavity-backed slot antenna radiates well because when the small dipole radiates close to the resonant frequency of the cavity, the field strength is strongly enhanced inside the cavity, and hence around the slot. This makes the slot into a good radiator (courtesy of antenna-theory.com). Slot antenna is a simple antenna to make [151]. To improve the radiation efficiency of slot Different Types of Antennas—Heuristics 307 antenna, it is made to radiate via a cavity. A cavity-backed slot antenna that uses such a concept is shown in Figure 28.4. A small dipole with poor radiation efficiency is placed inside the cavity. When the operating frequency is close to the resonant frequency of the cavity, the field strength inside the cavity becomes very strong, and much of the energy can leak from the cavity via the slot on its side. This makes the antenna radiate more efficiently into free space compared to just the small dipole alone. Another antenna that resembles a cavity backed slot antenna is the microstrip patch an- tenna, or patch antenna. This is shown in in Figure 28.5. This antenna also radiates efficiently by resonant tunneling. Roughly, when L (see left of Figure 28.5) is half a wavelength, the patch antenna resonates. This is similar to the resonant frequency of a transmission line with open circuit at both ends. The current sloshes back and forth across the length of the patch antenna along the L direction. The second design (right of Figure 28.5) has an inset feed. This allows the antenna to resonate at a lower frequency because the current has a longer path to slosh through when it is at resonance. Figure 28.5: A microstrip patch antenna also radiates well when it resonates. The patch antenna resembles a cavity resonator with magnetic wall (courtesy of emtalk.com). 28.2 Horn Antennas The impedance of free space is 377 ohms while that of most transmission line is 50 ohms. This impedance mismatch can be mitigated by using a flared horn (see Figure 28.6) [152]. One can think that the characteristic impedance of a transmission line made of two pieces p of metal as Z0 = L/C. As the horn flares, C becomes smaller, increasing its characteristic impedance to get close to that of free space. This allows for better impedance matching from the source to free space. This is similar to the quarter wave transformer for matching the characteristic impedance Z0 of a line to a load with impedance √ZL. The requirement is that the quarter wave transformer has an impedance given by ZT = Z0ZL A corrugated horn, as we have discussed previously in a circular waveguide in Section 20.1.1, discourages current flows in the non-axial symmetric mode. It encourages the prop- agation of the TE01 mode in the circular waveguide and hence, the circular horn antenna. This mode is axially symmetric, and thus, this antenna can radiate fields that are axially symmetric [153,154]. 308 Electromagnetic Field Theory Figure 28.6: A horn antenna works with the same principle as the biconical antenna. Its flared horn changes the waveguide impedance so as to match the impedance of a waveguide to the impedance of free space. The lower figure is that of a corrugated circular horn antenna. The corrugation enhances the propagation of the TE01 mode in the circular waveguide, and thus it enhances the cylindrical symmetry of the mode and the radiation field (courtesy of tutorialpoints.com and comsol.com). A Vivaldi antenna (invented by P. Gibson in 1978 [155]), is shown in Figure 28.7.2 It is also called a notch antenna. It works by the same principle to gradually match the impedance of the source to that of free space. But such a gradually flared horn has the element of a frequency independent antenna. The low frequency component of the signal will radiate from the wide end of the flared notch, while the high frequency component will radiate from the narrow end of the notch. Thus, this antenna can radiate effectively over a broad range of frequencies, and thus this gives the antenna a broad bandwidth performance. It is good for transmitting a pulsed signal which has a broad frequency spectrum. 2He must have loved the musician Vivaldi so much:) Different Types of Antennas—Heuristics 309 Figure 28.7: A Vivaldi antenna, also called a notch antenna, works like a horn antenna, but uses very little metal. Hence, it is cheap to build, and its flared notch makes it broadband (courtesy of Wikipedia [156]). 28.3 Quasi-Optical Antennas High-frequency or short wavelength electromagnetic field behaves like light ray as in optics. Therefore, many high-frequency antennas are designed based on the principle of ray optics. A reflector antenna is such an antenna as shown in Figure 28.8. The reflector antenna in this case is a Cassegrain design [157]3 where a sub-reflector is present. This allows the antenna to be fed from behind the parabolic dish where the electronics can be stored and isolated as well. Reflector antennas [159] are prevalent in radio astronomy and space exploration due to their high directivity and sensitivity. Figure 28.8: The left picture of an NRAO radio telescope antenna of Cassegrain design in Virginia, USA (courtesy of Britannica.com). The bottom is the detail of the Cassegrain design (courtesy of rev.com). 3The name came from an optical telescope of similar design [158] 310 Electromagnetic Field Theory Another recent invention is the reflectarray antenna [160,161] which is very popular. One of them is as shown in Figure 28.9. Due to recent advent in simulation technology, complicated structures can be simulated on a computer, including one with a complicated surface design. Patch elements can be etched onto a flat surface as shown, giving it an effective impedance that is spatially varying, making it reflect like a curved surface. Such a surface is known as a meta-surface [162, 163]. It can greatly economize on the space usage compared to a reflector antenna. Figure 28.9: A reflectarray where the reflector is a flat surface. Patches are unequally spaced to give the array the focussing effect (courtesy of antenna-theory.com). Another quasi-optical antenna is the lens antenna as shown in Figure 28.10 [164]. The design of this antenna follows lens optics, and is only valid when the wavelength is very short compared to the curvature of the surfaces. In this case, reflection and transmission at a curve surface is similar to that of a flat surface. This is called the tangent-plane approximation of a curve surface, and is valid at high frequencies. Different Types of Antennas—Heuristics 311 Figure 28.10: The left figure shows a lens antenna where the lens is made of artificial dielectrics made from metallic strips (courtesy of electriciantutoring.tpub.com). The right figure shows some dielectric lens at the aperture of an open waveguide to focus the microwave exiting from the waveguide opening (courtesy of micro-radar.de). 28.4 Small Antennas Small antennas are in vogue these days due to the advent of the cell phone, and the im- portance of economizing on the antenna size due to miniaturization requirements. Also, the antennas should have enough bandwidth to accommodate the signals from different cell phone companies, which use different carrier frequencies. An interesting small antenna is the PIFA (planar inverted F antenna) shown in Figure 28.11 [165]. Because it is shorted at one end and open circuit at the other end, it acts like a quarter wavelength resonator, making it substantially smaller. But the resonator has a low Q because of the “slots” or “openings” around it from whom energy can leak. The low Q gives this antenna a broader bandwidth. Figure 28.11: A PIFA (planar inverted F antenna) is compact, broadband, and easy to fabricate. It is good for cell phone antennas due to its small size (courtesy of Mathworks). 312 Electromagnetic Field Theory An interesting small antenna is the U-slot antenna shown in Figure 28.12 [166, 167]. Be- cause the current is forced to follow a longer tortuous path by the U-slot, it can resonant with a longer wavelength (lower frequency) and hence, can be made smaller compared to wave- length. In order to give the antenna a larger bandwidth, its Q is made smaller by etching it on a thick dielectric substrate (shown as the dielectric material region in the figure). But feeding it with a longer probe will make the bandwidth of the antenna smaller, due to the larger inductance of the probe.4 An ingenious invention is to use an L probe [168]. The L probe has an inductive part as well as a capacitive part. Their reactance cancel each other, allowing the electromagnetic energy to tunnel through the antenna, making it a better radiator. Figure 28.12: The top figure shows a U slot patch antenna design. The bottom figure shows a patch antenna fed by an L probe with significant increase in bandwidth (courtesy of K.M. Luk) [168]. Another area where small antennas are needed is in RFID (radio frequency identification) tag [169]. Since tags are placed outside the packages of products, e.g., in a warehouse, an RFID tag has a transmit-receive antenna that can communicate with the external world. The communication is done through an RFID reader. The RFID reader can talk to a small com- 4Remember that larger inductance implies more store magnetic field energy, and hence, the higher Q of the system. Different Types of Antennas—Heuristics 313 puter chip embedded in the tag where data about the package can be stored. Thus, an RFID reader can quickly and remotely communicate with the RFID tag to retrieve information about the package. Such a small antenna design for RFID tag is shown in Figure 28.13. It uses image theorem (that we shall learn later) so that the antenna can be made half as small. Then slots are cut into the radiating patch, so that the current follows a longer path. This lowers the resonant frequency of the antenna, allowing it to be made smaller. The take home message here is that to make an antenna a few times smaller than a wavelength to resonate, the current on the antenna has to flow through a tortuous path. In this manner, the antenna can be made a few times smaller than the wavelength. Figure 28.13: Some RFID antennas designed at The University of Hong Kong (courtesy of P. Yang, Y. Li, J. Huang, L.J. Jiang, S.Q. He, T. Ye, and W.C. Chew). An RFID reader can be designed to read the information from a batch of vials or test tubes containing different chemicals. Hence, a large loop antenna is needed but at a sufficiently high frequency (for large bandwidth). However, a loop antenna, if we look at a piece of wire as a Goubau line [145], will have resonant frequencies. When a loop antenna resonates, the current is non-uniform on it. This happens at higher frequencies. (Fundamentally, this comes from the retardation effect of electromagnetic field.) This will result in a non-uniform field inside the loop defeating the design of the RFID reader. One way to view how the non-uniform current come about is that a piece of wire becomes a tiny inductor. Across an inductor, V = jωLI, implying a 90◦ phase shift between the voltage and the current. In other words, the voltage drop is always nonzero, and therefore, the voltage cannot be constant around the loop. Since the voltage and current are locally 314 Electromagnetic Field Theory related by the local inductance, the current cannot be constant also. To solve the problem of the current and voltage being non-constant around the loop, a local inductor is connected in series with a capacitor [170]. This causes the local LC tank circuit to resonate. At resonance, the current-voltage relationship across the LC tank circuit is such that there is no voltage drop across the tank circuit since it becomes a short circuit. In this way, the voltage is equilized between two points and becomes uniform across the loop so is the current. Therefore, one way to enable a uniform current in a large loop is to capacitively load the loop. This will ensure a constant phase, or a more uniform current around the loop, and hence, a more efficient reader. Such a design is shown in Figure 28.14. Different Types of Antennas—Heuristics 315 Figure 28.14: The top figure shows a RFID reader designed by [171]. The bottom figure shows simulation and measurement done at The University of Hong Kong (courtesy of Z.N. Chen [171] and P. Yang, Y. Li, J. Huang, L.J. Jiang, S.Q. He, T. Ye, and W.C. Chew). 316 Electromagnetic Field Theory Lecture 29 Uniqueness Theorem The uniqueness of a solution to a linear system of equations is an important concept in mathematics. Under certain conditions, ordinary differential equation partial differential equation and matrix equations will have unique solutions. But uniqueness is not always guaranteed as we shall see. This issue is discussed in many math books and linear algebra books [77,89]. The proof of uniqueness for Laplace and Poisson equations are given in [31,54] which is slightly different from electrodynamic problems. Just imagine how bizzare it would be if there are more than one possible solutions. One has to determine which is the real solution. To quote Star Trek, we need to know who the real McCoy is!1 29.1 The Difference Solutions to Source-Free Maxwell’s Equations In this section, we will prove uniqueness theorem for electrodynamic problems [32, 35, 50, 65, 83]. First, let us assume that there exist two solutions in the presence of one set of common 2 a a b b impressed sources Ji and Mi. Namely, these two solutions are E , H , E , H . Both of them satisfy Maxwell’s equations and the same boundary conditions. Are Ea = Eb, Ha = Hb? To study the uniqueness theorem, we consider general linear anisotropic inhomogeneous media, where the tensors µ and ε can be complex so that lossy media can be included. In the frequency domain, lets assume two possible solutions with one given set of sources Ji and 1This phrase was made popular to the baby-boom generation, or the Trekkies by Star Trek. It actually refers to an African American inventor. 2It is not clear when the useful concept of impressed sources were first used in electromagnetics even though it was used in [172] in 1936. These are immutable sources that cannot be changed by the environment in which they are immersed. 317 318 Electromagnetic Field Theory Mi, it follows that a a ∇ × E = −jωµ · H − Mi (29.1.1) b b ∇ × E = −jωµ · H − Mi (29.1.2) a a ∇ × H = jωε · E + Ji (29.1.3) b b ∇ × H = jωε · E + Ji (29.1.4) By taking the difference of these two solutions, we have ∇ × (Ea − Eb) = −jωµ · (Ha − Hb) (29.1.5) ∇ × (Ha − Hb) = jωε · (Ea − Eb) (29.1.6) Or alternatively, defining δE = Ea − Eb and δH = Ha − Hb, we have ∇ × δE = −jωµ · δH (29.1.7) ∇ × δH = jωε · δE (29.1.8) The difference solutions, δE and δH, satisfy the original source-free Maxwell’s equations. Source-free here implies that we are looking at the homogeneous solutions of the pertinent partial differential equations constituted by (29.1.7) and (29.1.8). To prove uniqueness, we would like to find a simplifying expression for ∇ · (δE × δH∗). By using the product rule for divergence operator, it can be shown that ∇ · (δE × δH∗) = δH∗ · ∇ × δE − δE · ∇ × δH∗ (29.1.9) Then by taking the left dot product of δH∗ with (29.1.7), and then the left dot product of δE with the complex conjugation of (29.1.8), we obtain δH∗ · ∇ × δE = −jωδH∗ · µ · δH δE · ∇ × δH∗ = −jωδE · ε∗ · δE∗ (29.1.10) Now, taking the difference of the above, we get δH∗ · ∇ × δE − δE · ∇ × δH∗ = ∇ · (δE × δH∗) = −jωδH∗ · µ · δH + jωδE · ε∗ · δE∗ (29.1.11) Uniqueness Theorem 319 Figure 29.1: Geometry for proving the uniqueness theorem. We like to know the requisite boundary conditions on S plus the type of media inside V in order to guarantee the uniqueness of the solution in V . Next, we integrate the above equation (29.1.11) over a volume V bounded by a surface S as shown in Figure 29.1. After making use of Gauss’ divergence theorem, we arrive at ¤ ∇ · (δE × δH∗)dV = (δE × δH∗) · dS V ¦S = [−jωδH∗ · µ · δH + jωδE · ε∗ · δE∗]dV (29.1.12) V And next, we would like to know the kind of boundary conditions that would make the left-hand side equal to zero. It is seen that the surface integral on the left-hand side will be zero if:3 1. Ifn ˆ × E is specified over S for the two possible solutions, so thatn ˆ × Ea =n ˆ × Eb on S. Then n ˆ × δE = 0, which is the PEC boundary condition for δE, and then4 ∗ ∗ S(δE × δH ) · ndSˆ = S(ˆn × δE) · δH dS = 0. 2. Ifn ˆ × H is specified over S for the two possible solutions, so thatn ˆ × Ha =n ˆ × Hb on S. Then n ˆ × δH = 0, which is the PMC boundary condition for δH, and then ∗ ∗ S(δE × δH ) · ndSˆ = − S(ˆn × δH ) · δEdS = 0. 5 3. Let the surface S be divided into two mutually exclusive surfaces S1 and S2. Ifn ˆ × E is 3In the following, please be reminded that PEC stands for “perfect electric conductor”, while PMC stands for “perfect magnetic conductor”. PMC is the dual of PEC. Also, a fourth case of impedance boundary condition is possible, which is beyond the scope of this course. Interested readers may consult Chew, Theory of Microwave and Optical Waveguides [83]. 4We use the vector identity that a · (b × c) = c · (a × b) = b · (c × a). Also, from Section 1.3.3, dS =ndS ˆ . 5 In math parlance, S1 ∪ S2 = S. 320 Electromagnetic Field Theory specified over S1, andn ˆ ×H is specified over S2. Thenn ˆ ×δE = 0 (PEC boundary condition) on S1, andn ˆ × δH = 0 (PMC boundary condition) on S2. Therefore, the left-hand side becomes £ £ £ (δE × δH∗) · ndSˆ = + = (ˆn × δE) · δH∗dS S S1 S2 S1 £ − (ˆn × δH∗) · δEdS = 0. S2 Thus, under the above three scenarios, the left-hand side of (29.1.12) is zero, and then the right-hand side of (29.1.12) becomes ¦ [−jωδH∗ · µ · δH + jωδE · ε∗ · δE∗]dV = 0 (29.1.13) V For lossless media, µ and ε are hermitian tensors (or matrices6), then it can be seen, using the properties of hermitian matrices or tensors, that δH∗ · µ · δH and δE · ε∗ · δE∗ are purely real. Taking the imaginary part of the above equation yields ¦ [−δH∗ · µ · δH + δE · ε∗ · δE∗]dV = 0 (29.1.14) V The above two terms correspond to stored magnetic field energy and stored electric field energy in the difference solutions δH and δE, respectively. The above being zero does not imply that δH and δE are zero. For resonant solutions, the stored electric energy can balance the stored magnetic energy. The above resonant solutions are those of the difference solutions satisfying PEC or PMC boundary condition or mixture thereof. Also, they are the resonant solutions of the source-free Maxwell’s equations (29.1.7). Therefore, δH and δE need not be zero, even though (29.1.14) is zero. This happens when we encounter solutions that are the resonant modes of the volume V bounded by the surface S. 29.2 Conditions for Uniqueness Uniqueness can only be guaranteed if the medium is lossy as shall be shown later. It is also guaranteed if lossy impedance boundary conditions are imposed.7 First we begin with the isotropic case. 29.2.1 Isotropic Case It is easier to see this for lossy isotropic media. Then (29.1.13) simplifies to ¦ [−jωµ|δH|2 + jωε∗|δE|2]dV = 0 (29.2.1) V 6Tensors are a special kind of matrices. 7See Chew, Theory of Microwave and Optical Waveguides. Uniqueness Theorem 321 For isotropic lossy media, µ = µ0 − jµ00 and ε = ε0 − jε00. Taking the real part of the above, we have from (29.2.1) that ¦ [−ωµ00|δH|2 − ωε00|δE|2]dV = 0 (29.2.2) V Since the integrand in the above is always negative definite, the integral can be zero only if δE = 0, δH = 0 (29.2.3) everywhere in V , implying that Ea = Eb, and that Ha = Hb. Hence, it is seen that uniqueness is guaranteed only if the medium is lossy. The physical reason is that when the medium is lossy, a homogeneous solution (also called a natural solution) which is pure time-harmonic solution cannot exist due to loss. The modes, which are the source-free solutions of Maxwell’s equations, are decaying sinusoids. But when we express equations (29.1.1) to (29.1.4) in the frequency domain, we are seeking solutions for which ω is real. Thus decaying sinusoids are not among the possible solutions, and hence, they are not in the solution space. Notice that the same conclusion can be drawn if we make µ00 and ε00 negative. This corresponds to active media, and uniqueness can be guaranteed for a time-harmonic solution. In this case, no time-harmonic solution exists, and the resonant solution is a growing sinusoid. 29.2.2 General Anisotropic Case The proof for general anisotropic media is more complicated. For the lossless anisotropic media, we see that (29.1.13) is purely imaginary. However, when the medium is lossy, this same equation will have a real part. Hence, we need to find the real part of (29.1.13) for the general lossy case. About taking the Real and Imaginary Parts of a Complicated Expression To this end, we digress on taking the real and imaginary parts of a complicated expression. Here, we need to find the complex conjugate8 of (29.1.13), which is scalar, and add it to itself to get its real part. To this end, we will find the conjugate of its integrand which is a scalar number. First, the complex conjugate of the first scalar term in the integrand of (29.1.13) is9 (−jωδH∗ · µ · δH)∗ = jωδH · µ∗ · δH∗ = jωδH∗ · µ† · δH (29.2.4) Similarly, the complex conjugate of the second scalar term in the same integrand is (jωδE · ε∗ · δE∗)∗ = −jωδE∗ · ε† · δE (29.2.5) 8Also called hermitian conjugate. 9To arrive at these expressions, one makes use of the matrix algebra rule that if D = A · B · C, then t t t t D = C · B · A . This is true even for non-square matrices. But for our case here, A is a 1 × 3 row vector, and C is a 3×1 column vector, and B is a 3×3 matrix. In vector algebra, the transpose of a vector is implied. Also, in our case here, D is a scalar, and hence, its transpose is itself. 322 Electromagnetic Field Theory But jωδE · ε∗ · δE∗ = jωδE∗ · ε† · δE (29.2.6) The above gives us the complex conjugate of the scalar quantity (29.1.13) and adding it to itself, we have ¦ [−jωδH∗ · (µ − µ†) · δH − jωδE∗ · (ε − ε†) · δE]dV = 0 (29.2.7) V For lossy media, −j(µ − µ†) and −j(ε − ε†) are hermitian positive matrices. Hence the integrand is always positive definite, and the above equation cannot be satisfied unless δH = δE = 0 everywhere in V . Thus, uniqueness is guaranteed in a lossy anisotropic medium. Similar statement can be made for the isotropic case if the medium is active. Then the integrand is positive definite, and the above equation cannot be satisfied unless δH = δE = 0 everywhere in V , thereby proving that uniqueness is satisfied. 29.3 Hind Sight Using Linear Algebra The proof of uniqueness for Maxwell’s equations is very similar to the proof of uniqueness for a matrix equation [77]. As you will see, the proof using linear algebra is a lot simpler due to the simplicity of notations. To see this, consider a linear algebraic equation A · x = b (29.3.1) If a solution to a matrix equation exists without excitation, namely, when b = 0, then the solution is the null space solution [77], namely, x = xN . In other words, A · xN = 0 (29.3.2) These null space solutions exist without a “driving term” b on the right-hand side. For Maxwell’s equations, b corresponds to the source terms. The solution in (29.3.2) is like the homogeneous solution of an ordinary differential equation or a partial differential equation [89]. In an enclosed region of volume V bounded by a surface S, homogeneous solutions are the resonant solutions (or the natural solutions) of this Maxwellian system. When these solutions exist, they give rise to non-uniqueness. Note that these resonant solutions in the time domain exist for all time if the cavity is lossless. Also, notice that (29.1.7) and (29.1.8) are Maxwell’s equations without the source terms. In a closed region V bounded by a surface S, only resonant solutions for δE and δH with the relevant boundary conditions can exist when there are no source terms. As previously mentioned, one way to ensure that these resonant solutions (or homoge- neous solutions) are eliminated is to put in loss or gain. When loss or gain is present, then the resonant solutions are decaying sinusoids or growing sinusoids (see Section 22.1.1 for an analogue with LC tank circuit). Since we are looking for solutions in the frequency domain, or time harmonic solutions, the solutions considered are on the real ω axis on the complex ω plane. Thus the non-sinusoidal solutions are outside the solution space: They are not part of the time-harmonic solutions (which are on the real axis) that we are looking for. Therefore, Uniqueness Theorem 323 complex resonant solutions which are off the real axis, and are homogeneous solutions, are not found on the real axis. We see that the source of non-uniqueness is the homogeneous solutions or the resonant solutions of the system that persist for all time. These solutions are non-causal, and they are there in the system since the beginning of time to time ad infinitum. One way to remove these resonant solutions is to set them to zero at the beginning by solving an initial value problem (IVP). However, this has to be done in the time domain. One reason for non-uniqueness is because we are seeking the solutions in the frequency domain. 29.4 Connection to Poles of a Linear System The output to input of a linear system can be represented by a transfer function H(ω) [52,173]. If H(ω) has poles, and if the system is lossless, the poles are on the real axis. Therefore, when ω = ωpole, the function H(ω) becomes undefined. In other words, one can add a constant term to the output, and the ratio between output to input is still infinity. This also gives rise to non-uniqueness of the output with respect to the input. Poles usually correspond to resonant solutions, and hence, the non-uniqueness of the solution is intimately related to the non-uniqueness of Maxwell’s equations at the resonant frequencies of a structure. This is illustrated in the upper part of Figure 29.2. Figure 29.2: The non-uniqueness problem is intimately related to the locations of the poles of a transfer function being on the real axis, when one solves a linear system using Fourier transform technique. For a lossless system, the poles are located on the real axis. When performing a Fourier inverse transform to obtain the solution in the time domain, then the Fourier inversion contour is undefined, and the solution cannot be uniquely determined. 324 Electromagnetic Field Theory If the input function is f(t), with Fourier transform F (ω), then the output y(t) is given by the following Fourier integral, viz., ¢ 1 ∞ y(t) = dωejωtH(ω)F (ω) (29.4.1) 2π −∞ where the Fourier inversion integral path is on the real axis on the complex ω plane. The Fourier inversion integral is undefined or non-unique if poles exist on the real ω axis. However, if loss is introduced, these poles will move away from the real axis as shown in the lower part of Figure 29.2. Then the transfer function is uniquely determined for all frequencies on the real axis. In this way, the Fourier inversion integral in (29.4.1) is well defined, and uniqueness of the solution is guaranteed. When the poles are located on the real axis yielding possibly non-unique solutions, a remedy to this problem is to use Laplace transform technique [52]. The Laplace transform technique allows the specification of initial values, which is similar to solving the problem as an initial value problem (IVP). If you have problem wrapping your head around this concept, it is good to connect back to the LC tank circuit example. The transfer function H(ω) is similar to the Y (ω) of (22.1.4). The transfer function has two poles. If there is no loss, then the poles are located on the real axis, rendering the Fourier inversion contour undefined in (29.4.1). Hence, the solution is non-unique. However, if infinitesimal loss is introduced by setting R 6= 0, then the poles will migrate off the real axis making (29.4.1) well defined! 29.5 Radiation from Antenna Sources and Radiation Con- dition The above uniqueness theorem guarantees that if we have some antennas with prescribed current sources on them, the radiated field from these antennas are unique. To see how this can come about, we first study the radiation of sources into a region V bounded by a large surface Sinf as shown in Figure 29.4 [35]. Even whenn ˆ × E orn ˆ × H are specified on the surface at Sinf, the solution is nonunique because the volume V bounded by Sinf, can have many resonant solutions. In fact, the region will be replete with resonant solutions as one makes Sinf become very large. To have more insight, we look at the resonant condition of a large rectangular cavity given by (21.2.3) reproduced here as ω2 mπ 2 nπ 2 pπ 2 β2 = = + + (29.5.1) c2 a b d mπ The above is an equation of an Ewald sphere in a 3D mode space, but the values of βx = a , nπ pπ βy = b , and βz = d are discrete. We can continuously change the operating frequency ω above until the above equation is satisfied. When this happens, we encounter a resonant frequency of the cavity. At this operating frequency, the solution to Maxwell’s equations inside the cavity is non-unique. As the dimensions of the cavity become large or a, b, and d are large, then the number of ω’s or resonant frequencies that the above equation can be Uniqueness Theorem 325 satisfied or approximately satisfied becomes very large. This is illustrated Figure 29.3 in 2D. Hence, the chance of the operating frequency ω coinciding with a resonant mode of the cavity is very high giving rise to nonuniqueness. This is even more so when the cavity becomes very large. Hence, the chance of operating inside the cavity with unique solution is increasingly difficult. This above argument applies to cavities of other shapes as well. Figure 29.3: For very large cavity, the grid spacing in the mode space (or Fourier space) becomes very small. Then the chance that the sphere surface encounters a resonant mode is very high. When this happens, the solution to the cavity problem is non-unique. The way to remove these resonant solutions is to introduce an infinitesimal amount of loss in region V . Then these resonant solutions will disappear from the real ω axis, where we seek a time-harmonic solution. Now we can take Sinf to infinity, and the solution will always be unique even if the loss is infinitesimally small. Notice that if Sinf → ∞, the waves that leave the sources will never be reflected back because of the small amount of loss. The radiated field will just disappear into infinity. This is just what radiation loss is: power that propagates to infinity, but never to return. In fact, one way of guaranteeing the uniqueness of the solution in region V when Sinf is infinitely large, or that V is infinitely large is to impose the radiation condition: the waves that radiate to infinity are outgoing waves only, and never do they return. This is also called the Sommerfeld radiation condition [174]. . Uniqueness of the field outside the sources is always guaranteed if we assume that the field radiates to infinity and never to return. This is equivalent to solving the cavity solutions with an infinitesimal loss, and then letting the size of the cavity become infinitely large. 326 Electromagnetic Field Theory Figure 29.4: The solution for antenna radiation is unique because we impose the Sommerfeld radiation condition when seeking the solution. That is we assume that the radiation wave travels to infinity but never to return. This is equivalent to assuming an infinitesimal loss when seeking the solution in V . Lecture 30 Reciprocity Theorem Reciprocity theorem is one of the most important theorems in electromagnetics. With it we can develop physical intuition to ascertain if a certain design or experiment is right or wrong. It also tells us what is possible or impossible in the design of many systems. Reciprocity theorem is like “tit-for-tat” relationship in humans: Good-will is reciprocated with good will while ill-will is countered with ill-will. Both Confucius (551 BC–479 BC) and Jesus Christ (4 BC–AD 30) epoused the concept that, “Don’t do unto others that you don’t like others to do unto you.” But in electromagnetics, this beautiful relationship can be expressed precisely and succinctly using mathematics. We shall see how this is done. Figure 30.1: (Left) A depiction of Confucius from a stone fresco from the Western Han dynasty (202 BC–9 AD). The emphasis of the importance of “reciprocity” by Confucius Analects translated by D. Hinton [175]. (Right) A portrait of Jesus that is truer to its form. Jesus teaching from the New Testament says, “Do unto others as you would have them do unto you.” Luke 6:31 and Matthew 7:12 [176]. The subsequent portraits of these two sages are more humanly urbane. 327 328 Electromagnetic Field Theory 30.1 Mathematical Derivation Figure 30.2: The geometry for proving reciprocity theorem. We perform two experiments on the same object or scatterer: (a) With sources J1 and M1 turned on, generating fields E1 and H1, and (b) With sources J2 and M2 turned on, generating fields E2 and H2. Magnetic currents, by convention, are denoted by double arrows. Consider a general anisotropic inhomogeneous medium in the frequency domain where both µ(r) and ε(r) are described by permeability tensor and permittivity tensor over a finite part of space as shown in Figure 30.2. This representation of the medium is quite general, and it can include dispersive and conductive media as well. It can represent complex terrain, or complicated electronic circuit structures in circuit boards or microchips, as well as complicated antenna structures. We can do a Gedanken experiment1 where a scatterer or an object is illuminated by fields from two sets of sources which are turned on and off consecutively. This is illustrated in Figure 30.2: When only J1 and M1 are turned on, they generate fields E1 and H1 in this medium. On the other hand, when only J2 and M2 are turned on, they generate E2 and H2 in this medium. Therefore, the pertinent equations in the frequency domain, for linear time-invariant systems, for these two cases are2 ∇ × E1 = −jωµ · H1 − M1 (30.1.1) ∇ × H1 = jωε · E1 + J1 (30.1.2) ∇ × E2 = −jωµ · H2 − M2 (30.1.3) ∇ × H2 = jωε · E2 + J2 (30.1.4) We would like to find a simplifying expression for the divergence of the following quantity, ∇ · (E1 × H2) = H2 · ∇ × E1 − E1 · ∇ · H2 (30.1.5) so that the divergence theorem can be invoked. To this end, and from the above, we can show 1Thought experiment in German. 2The current sources are impressed currents so that they are immutable, and not changed by the environ- ment they are immersed in [50, 172]. Reciprocity Theorem 329 that (after left dot-multiply (30.1.1) with H2 and (30.1.4) with E1), H2 · ∇ × E1 = −jωH2 · µ · H1 − H2 · M1 (30.1.6) E1 · ∇ × H2 = jωE1 · ε · E2 + E1 · J2 (30.1.7) Then, using the above, and the following identity, we get the second equality in the following expression: ∇ · (E1 × H2) = H2 · ∇ × E1 − E1 · ∇ × H2 = −jωH2 · µ · H1 − jωE1 · ε · E2 − H2 · M1 − E1 · J2 (30.1.8) By the same token, ∇ · (E2 × H1) = −jωH1 · µ · H2 − jωE2 · ε · E1 − H1 · M2 − E2 · J1 (30.1.9) If one assumes that µ = µt, ε = εt (30.1.10) 3 or when the tensors are symmetric, then H1 · µ · H2 = H2 · µ · H1 and E1 · ε · E2 = E2 · ε · E1. Upon subtracting (30.1.8) and (30.1.9), one gets ∇ · (E1 × H2 − E2 × H1) = −H2 · M1 − E1 · J2 + H1 · M2 + E2 · J1 (30.1.11) Figure 30.3: The geometry for proving reciprocity theorem when the surface S: (a) does not enclose the sources, and (b) encloses the sources. In the figure, the sources are supposed to be either (M1, J1) producing fields (E1, H1) or (M2, J2) producing fields (E2, H2). Now, integrating (30.1.11) over a volume V bounded by a surface S, and invoking Gauss’ divergence theorem, we have the reciprocity theorem that dS · (E1 × H2 − E2 × H1) S ¦ = − dV [H2 · M1 + E1 · J2 − H1 · M2 − E2 · J1] (30.1.12) V 3It is to be noted that in matrix algebra, the dot product between two vectors are often written as at · b, but in the physics literature, the transpose on a is implied. Therefore, the dot product between two vectors is just written as a · b. 330 Electromagnetic Field Theory When the volume V contains no sources (see Figure 30.3), the reciprocity theorem reduces to dS · (E1 × H2 − E2 × H1) = 0 (30.1.13) S The above is also called Lorentz reciprocity theorem by some authors.4 Next, when the surface S contains all the sources (see Figure 30.3), then the right-hand side of (30.1.12) will not be zero. On the other hand, when the surface S → ∞, E1 and H2 becomes spherical waves which can be approximated by plane waves sharing the same β vector. Moreover, under the plane-wave approximation, ωµ0H2 = β × E2, ωµ0H1 = β × E1, then E1 × H2 ∼ E1 × (β × E2) = E1(β · E2) − β(E1 · E2) (30.1.14) E2 × H1 ∼ E2 × (β × E1) = E2(β · E1) − β(E2 · E1) (30.1.15) But β ·E2 = β ·E1 = 0 in the far field and the β vectors are parallel to each other. Therefore, the two terms on the left-hand side of (30.1.12) cancel each other, and it vanishes when S → ∞. (They cancel each other so that the remnant field vanishes faster than 1/r2. This is necessary as the surface area S is growing larger and proportional to r2.) As a result, when S → ∞, (30.1.12) can be rewritten simply as ¢ ¢ dV [E2 · J1 − H2 · M1] = dV [E1 · J2 − H1 · M2] (30.1.16) V V The inner product symbol is often used to rewrite the above as hE2, J1i − hH2, M1i = hE1, J2i − hH1, M2i (30.1.17) ¡ where the inner product hA, Bi = V dV A(r) · B(r). The above inner product is also called reaction, a concept introduced by Rumsey [177]. The above is also called the Rumsey reaction theorem. Sometimes, the above is rewritten more succinctly and tersely as h2, 1i = h1, 2i (30.1.18) where h2, 1i = hE2, J1i − hH2, M1i (30.1.19) The concept of inner product or reaction can be thought of as a kind of “measurement”. The reciprocity theorem can be stated as that the fields generated by sources 2 as “measured” by sources 1 is equal to fields generated by sources 1 as “measured” by sources 2. This measurement concept is more lucid if we think of these sources as Hertzian dipoles. 4Harrington, Time-Harmonic Electric Field [50]. Reciprocity Theorem 331 30.2 Conditions for Reciprocity It is seen that the above proof hinges on (30.1.10). In other words, the anisotropic medium has to be described by symmetric tensors. They include conductive media, but not gyrotropic media which is non-reciprocal. A ferrite biased by a magnetic field is often used in electronic circuits, and it corresponds to a gyrotropic, non-reciprocal medium.5 Also, our starting equations (30.1.1) to (30.1.4) assume that the medium and the equations are linear time invariant so that Maxwell’s equations can be written down in the frequency domain easily. 30.3 Application to a Two-Port Network and Circuit Theory Figure 30.4: A geometry for proving the circuit relationship between two antennas using reciprocity theorem. Circuit relationship is possible when the ports of the antennas are small compared to wavelength. The reciprocity theorem can be used to distill and condense the interaction between two antennas over a complex terrain as long as the terrain comprises reciprocal media, namely, if µ = µt and ε = εt for these media.6 In Figure 30.4, we assume that antenna 1 is driven by impressed current J1 while antenna 2 is driven by impressed current J2. It is assumed that the antennas are made from reciprocal media, such as conductive media. Since the system is linear time invariant, it can be written as the interaction between two ports as in circuit theory as shown in Figure 30.5. Assuming that these two ports are small compared to wavelengths, then we can apply circuit concepts like potential theory by letting E = −∇Φ in the neighborhood of the ports. Thus, we can define voltages and currents at these ports, and V-I relationships can be established in the manner of circuit theory. 5Non-reciprocal media are important for making isolators in microwave. Microwave signals can travel from Port 1 to Port 2, but not vice versa. 6It is to be noted that a gyrotropic medium considred in Section 9.1 does not safisfy this reciprocity criteria, but it does satisfy the lossless criteria of Section 10.3.2. 332 Electromagnetic Field Theory Figure 30.5: The interaction between two antennas in the far field of each other can be reduced to a circuit theory description since the input and output ports of the antennas are small compared to wavelength. Focusing on a two-port network as shown in Figure 30.5, we have from circuit theory that [178] V Z Z I 1 = 11 12 1 (30.3.1) V2 Z21 Z22 I2 This form is permissible since we have a linear time-invariant system, and this is the most general way to establish a linear relationship between the voltages and the currents. Further- more, the matrix elements Zij can be obtained by performing a series of open-circuit and short-circuit measurements as in circuit theory. Then assuming that the port 2 is turned on with J2 6= 0, while port 1 is turned off with J1 = 0. In other words, port 1 is open circuit, and the source J2 is an impressed current 7 source that will produce an electric field E2 at port 1. Since the current at port 1 is turned oc off, or that J1 = 0, the voltage measured at port 1 is the open-circuit voltage V1 . Please note here that J1 and J2 are impressed currents and are only defined in their respective port. Consequently, the reaction ¢ ¢ oc hE2, J1i = dV (E2 · J1) = I1 E2 · dl = −I1V1 (30.3.2) V Port 1 Even though port 1 is assumed to be off, the J1 is the impressed current to be used above is the J1 when port 1 is turned on. Given that the port is in the circuit physics regime, we assume the currents in wire to be constant, then the current J1 is a constant current at the port when it is turned on. Or the current I1 can be taken outside the integral. In slightly ˆ ˆ more details, the current J1 = lI1/A where A is the cross-sectional area of the wire, and l is a unit vector aligned with the axis of the wire. The volume¡ integral dV = Adl, and hence the ˆ oc second equality follows above, where dl = ldl. Since Port 1 E2 · dl = −V1 , we have the last equality above. We can repeat the derivation with port 2 to arrive at the reaction ¢ oc hE1, J2i = I2 E1 · dl = −I2V2 (30.3.3) Port 2 7This is the same as the current source concept in circuit theory. Reciprocity Theorem 333 Reciprocity requires these two reactions to be equal, and hence, oc oc I1V1 = I2V2 But from (30.3.1), we can set the pertinent currents to zero to find these open circuit voltages. oc oc oc oc Therefore, V1 = Z12I2, V2 = Z21I1. Since I1V1 = I2V2 by the reaction concept or by reciprocity, then Z12 = Z21. The above analysis can be easily generalized to an N-port network. The simplicity of the above belies its importance. The above shows that the reciprocity concept in circuit theory is a special case of reciprocity theorem for electromagnetic theory. The terrain can also be replaced by complex circuits as in a circuit board, as long as the materials in the terrain or circuit board are reciprocal, linear and time invariant. For instance, the complex terrain can also be replaced by complex antenna structures. It is to be noted that even when the transmit and receive antennas area miles apart, as long as the transmit and receive ports of the linear system can be characterized by a linear relation expounded by (30.3.1), and the ports small enough compared to wavelength so that circuit theory prevails, we can apply the above analysis! This relation that Z12 = Z21 is true as long as the medium traversed by the fields is a reciprocal medium. Before we conclude this section, it is to be mentioned that some researchers advocate the use of circuit theory to describe electromagnetic theory. Such is the case in the transmission line matrix (TLM) method [179], and the partial element equivalence circuit (PEEC) method [180]. Circuit theory is so simple that many people fall in love with it! 30.4 Voltage Sources in Electromagnetics Figure 30.6: Two ways to model voltage sources: (i) An impressed current source Ja driving a very short antenna, and (ii) An impressed magnetic frill source (loop source) Ma driving a very short antenna (courtesy of Kong, Electromagnetic Wave Theory [32]). In the above discussions, we have used current sources in reciprocity theorem to derive certain circuit concepts. Before we end this section, it is prudent to mention how voltage sources are 334 Electromagnetic Field Theory modeled in electromagnetic theory. The use of the impressed currents so that circuit concepts can be applied is shown in Figure 30.6. The antenna in (a) is driven by a current source. But a magnetic current can be used as a voltage source in circuit theory as shown by Figure 30.6b. By using duality concept, an electric field has to curl around a magnetic current just in Ampere’s law where magnetic field curls around an electric current. This electric field will cause a voltage drop between the metal above and below the magnetic current loop making it behave like a voltage source.8 30.5 Hind Sight The proof of reciprocity theorem for Maxwell’s equations is very deeply related to the sym- metry of the operator involved. We can see this from linear algebra. Given a matrix equation driven by two different sources b1 and b2 with solutions x1 and x2, they can be written succinctly as A · x1 = b1 (30.5.1) A · x2 = b2 (30.5.2) We can left dot multiply the first equation with x2 and do the same with the second equation with x1 to arrive at t t x2 · A · x1 = x2 · b1 (30.5.3) t t x1 · A · x2 = x1 · b2 (30.5.4) If A is symmetric, the left-hand side of both equations are equal to each other.9 Therefore, we can equate their right-hand side to arrive at t t x2 · b1 = x1 · b2 (30.5.5) The above is analogous to the statement of the reciprocity theorem which is hE2, J1i = hE1, J2i (30.5.6) ¡ where the reaction inner product, as mentioned before, is hEi, Jji = V Ei(r)·Jj(r). The inner product in linear algebra is that of dot product in matrix theory, but the inner product for reciprocity theorem is that for infinite dimensional spaces.10 So if the operators in Maxwell’s equations are symmetrical, then reciprocity theorem applies. 8More can be found in Jordain and Balmain, Electromagnetic Waves and Radiation Systems [54]. 9This can be easily proven by taking the transpose of a scalar, and taking the transpose of the product of matrices. 10Such spaces are called Hilbert space. Reciprocity Theorem 335 30.6 Transmit and Receive Patterns of an Antennna Figure 30.7: The schematic diagram for studying the transmit and receive properties of antennas. The two antennas are assumed to be identical, and each switches between transmit and receive modes in this study. Reciprocity also implies that the transmit and receive properties of an antenna is similar to each other. The transmit property of an antenna is governed by the gain function, while its receive property is governed by the effective area or aperture. The effective aperture is also a function of angle of the incident wave with respect to to the antenna. The gain function of an antenna is related to its effective aperture by a constant as we shall argue. Consider an antenna in the transmit mode. Then the time-average radiation power density that it will yield around the antenna, in accordance to (25.3.5), is11 P hS i = t G(θ, φ) (30.6.1) rad 4πr2 where Pt is the total power radiated by the transmit antenna,¡ and G(θ, φ) is its directive gain pattern or function. It is to be noted that in the above 4π dΩG(θ, φ) = 4π. The above is valid when the antenna is lossless. Effective Gain versus Directive Gain At this juncture, it is important to introduce the concept of effective gain versus directive gain. The effective gain, also called the power gain, is Ge(θ, φ) = feG(θ, φ) (30.6.2) where fe is the efficiency of the antenna, a factor less than 1. It accounts for the fact that not all power pumped into the antenna is delivered as radiated power. For instance, power 11The author is indebted to inspiration from E. Kudeki of UIUC for this part of the lecture notes [139]. 336 Electromagnetic Field Theory can be lost in the circuits and mismatch of the antenna. Therefore, the correct formula the radiated power density is P P hS i = t G (θ, φ) = f t G(θ, φ) (30.6.3) rad 4πr2 e e 4πr2 This radiated power resembles that of a plane wave when one is far away from the trans- mitter. Thus if a receive antenna is placed in the far-field of the transmit antenna, it will see this power density as coming from a plane wave. Thus the receive antenna will see an incident power density as P hS i = hS i = t G (θ, φ) (30.6.4) inc rad 4πr2 e Effective Aperture The effective area or the aperture of a receive antenna is used to characterize its receive property. The power received by such an antenna is then, by using the concept of effective aperture expounded in (26.1.23) 0 0 Pr = hSinciAe(θ , φ ) (30.6.5) where (θ0, φ0) are the angles at which the plane wave is incident upon the receiving antenna (see Figure 30.7). Combining the above formulas (30.6.4) and (30.6.5), we have P P = t G (θ, φ)A (θ0, φ0) (30.6.6) r 4πr2 e e Now assuming that the transmit and receive antennas are identical. Next, we swap their roles of transmit and receive, and also the circuitries involved in driving the transmit and receive antennas. Then, P P = t G (θ0, φ0)A (θ, φ) (30.6.7) r 4πr2 e e We also assume that the receive antenna, that now acts as the transmit antenna is transmitting in the (θ0, φ0) direction. Moreover, the transmit antenna, that now acts as the receive antenna is receiving in the (θ, φ) direction (see Figure 30.7). By reciprocity, these two powers are the same, because Z12 = Z21. Furthermore, since these two antennas are identical, Z11 = Z22. So by swapping the transmit and receive electronics, the power transmitted and received will not change. A simple transmit-receive circuit diagram is shown in Figure 30.8 Reciprocity Theorem 337 Figure 30.8: The schematic of the circuit for a transmit-receive antenna pair. Because the mutual interaction between the two antennas can be described by the impedance matrix Z, circuit theory can be applied to model their mutual interaction as indicated in (a). Moreover, at the receive end, one can even further simplify the circuit by using a Thevenin equivalence (b), or a Norton equivalence (c). Consequently, we conclude that 0 0 0 0 Ge(θ, φ)Ae(θ , φ ) = Ge(θ , φ )Ae(θ, φ) (30.6.8) The above implies that 0 0 Ae(θ, φ) Ae(θ , φ ) = 0 0 = constant (30.6.9) Ge(θ, φ) Ge(θ , φ ) The above Gedanken experiment is carried out for arbitrary angles. Therefore, the constant is independent of angles. Moreover, this constant is independent of the size, shape, and efficiency of the antenna, as we have not stipulated their shapes, sizes, and efficiency in the above discussion. To find this constant in (30.6.9), one can repeat the above for a Hertzian dipole, wherein the mathematics of calculating Pr and Pt is a lot simpler. This constant is found to be 338 Electromagnetic Field Theory λ2/(4π).12 Therefore, an interesting relationship between the effective aperture (or area) and the directive gain function is that λ2 A (θ, φ) = G (θ, φ) (30.6.10) e 4π e One amusing point about the above formula is that the effective aperture, say of a Hertzian dipole, becomes very large when the frequency is low, or the wavelength is very long. Of course, this cannot be physically true, and I will let you meditate on this paradox and muse over this point. 12See Kong [32][p. 700]. The derivation is for 100% efficient antenna. A thermal equilibrium argument is used in [139] and Wikipedia [140] as well. Lecture 31 Equivalence Theorems, Huygens’ Principle Electromagnetic equivalence theorems are useful for simplifying solutions to many problems. Also, they offer physical insight into the behaviour of electromagnetic fields of a Maxwellian system. They are closely related to the uniqueness theorem and Huygens’ principle. One application is their use in studying the radiation from an aperture antenna or from the output of a lasing cavity. These theorems are discussed in many textbooks [32,50,54,65,181]. Some authors also call it Love’s equivalence principles [182] and credit has been given to Schelkunoff as well [172]. You may have heard of another equivalence theorem in special relativity. It was postulated by Einstein to explain why light ray should bend around a star. The equivalence theorem in special relativity is very different from those in electromagnetics. One thing they have in common is that they are all derived by using Gedanken experiment (thought experiment) , involving no math. But in this lecture, we will show that electromagnetic equivalence theorems are also derivable using mathematics, albeit with more work. 31.1 Equivalence Theorems or Equivalence Principles In this lecture, we will consider three equivalence theorems: (1) The inside out case. (2) The outside in case. (3) The general case. As mentioned above, we will derive these theorems using thought experiments or Gedanken experiments. As shall be shown later, they can also be derived mathematically using Green’s theorem. 339 340 Electromagnetic Field Theory 31.1.1 Inside-Out Case Figure 31.1: The inside-out problem where the two cases in (a) and (b) are equivalent. In (b), equivalence currents are impressed on the surface S so as to produce the same fields outside in Vo in both cases, cases (a) and (b). In this case, we let J and M be the time-harmonic radiating sources inside a surface S radiating into a region V = Vo ∪ Vi. They produce E and H everywhere. We can construct an equivalence problem by first constructing an imaginary surface S. In this equivalence problem, the fields outside S in Vo are the same in both (a) and (b). But in (b), the fields inside S in Vi are zero. Apparently, in case (b) in Figure 31.1, the tangential components of the fields are discon- tinuous at S. This is not possible for a Maxwellian fields unless surface currents are impressed on the surface S. We have learned from electromagnetic boundary conditions that electro- magnetic fields are discontinuous across a current sheet. Then we ask ourselves what surface currents are needed on surface S so that the boundary conditions for field discontinuities are satisfied on S. Clearly, surface currents needed for these field discontinuities. By virtue of the boundary conditions and the jump conditions in electromagnetics, these surface currents to be impressed on S are Js =n ˆ × H, Ms = E × nˆ (31.1.1) We have learnt from Section 4.3.3 that an electric current sheet in Ampere’s law produced a jump discontinuity in the magnetic field. By the same token, when fictitious magnetic current is added to Faraday’s law in Section 5.3 for mathematical symmetry, a magnetic current sheet induces a jump discontinuity in the electric field. Because of the opposite polarity of the magnetic current M in Faraday’s law compared to the electric current in Ampere’s law as is shown in Section 5.3, this magnetic current sheet is proportional to E × nˆ instead of ׈ H. Consequetly, we can convince ourselves thatn ˆ × H and E × nˆ just outside S in both cases are the same. Furthermore, we are persuaded that the above is a bona fide solution to Maxwell’s equations. Equivalence Theorems, Huygens’ Principle 341 The boundary conditions on the surface S satisfy the boundary conditions required of Maxwell’s equations. By the uniqueness theorem, only the equality of one of them E × nˆ, orn ˆ × H ons S, will guarantee that E and H outside S are the same in both cases (a) and (b). The fact that these equivalence currents generate zero fields inside S is known as the extinction theorem. This equivalence theorem can also be proved mathematically, as shall be shown. 31.1.2 Outside-in Case Figure 31.2: The outside-in problem where equivalence currents are impressed on the surface S to produce the same fields inside in both cases. Similar to before, we find an equivalence problem (b) where the fields inside S in Vi is the same as in (a), but the fields outside S in Vo in the equivalence problem is zero. The fields are discontinuous across the surface S, and hence, impressed surface currents are needed to account for these discontinuities. 1 Then by the uniqueness theorem, the fields Ei, Hi inside V in both cases are the same. Again, by the extinction theorem, the fields produced by Ei × nˆ andn ˆ × Hi are zero outside S. 31.1.3 General Case From these two cases, we can create a rich variety of equivalence problems. By linear su- perposition of the inside-out problem, and the outside-in problem, then a third equivalence problem is shown in Figure 31.3: 1 We can add infinitesimal loss to ensure that uniqueness theorem is satisfied in this enclosed volume Vi. 342 Electromagnetic Field Theory Figure 31.3: The general case where the fields are non-zero both inside and outside the surface S. Equivalence currents are needed on the surface S to support the jump discontinuities in the fields. 31.2 Electric Current on a PEC Using the equivalence problems in the previous section, we can derive other corollaries of equivalence theorems. We shall show them next. First, from reciprocity theorem, it is quite easy to prove that an impressed current on the PEC cannot radiate. We can start with the inside-out equivalence problem. Since the fields inside S is zero for the inside-out problem, using a Gedanken experiment, one can insert an PEC object inside S without disturbing the fields E and H outside since the field is zero inside S. As the PEC object grows to snugly fit the surface S, then the electric current Js =n ˆ ×H does not radiate by reciprocity. Only one of the two currents is radiating, namely, the magnetic current Ms = E × nˆ is radiating, and Js in Figure 31.4 can be removed. This is commensurate with the uniqueness theorem that only the knowledge of E × nˆ is needed to uniquely determine the fields outside S. Figure 31.4: On a PEC surface, only one of the two currents is needed since an electric current does not radiate on a PEC surface. Equivalence Theorems, Huygens’ Principle 343 31.3 Magnetic Current on a PMC Again, from reciprocity, an impressed magnetic current on a PMC cannot radiate. By the same token, we can perform the Gedanken experiment as before by inserting a PMC object inside S. It will not alter the fields outside S, as the fields inside S is zero. As the PMC object grows to snugly fit the surface S, only the electric current Js =n ˆ × H radiates, and the magnetic current Ms = E × nˆ does not radiate and it can be removed. This is again commensurate with the uniqueness theorem that only the knowledge of then ˆ × H is needed to uniquely determine the fields outside S. Figure 31.5: Similarly, on a PMC surface, only an electric current is needed to produce the field outside the surface S. 31.4 Huygens’ Principle and Green’s Theorem Huygens’ principle shows how a wave field on a surface determines the wave field outside the surface S. This concept was expressed by Huygens heuristically in the 1600s [183]. But the mathematical expression of this idea was due to George Green 2 in the 1800s. This concept can be expressed mathematically for both scalar and vector waves. The derivation for the vector wave case is homomorphic to the scalar wave case. But the algebra in the scalar wave case is much simpler. Therefore, we shall discuss the scalar wave case first, followed by the electromagnetic vector wave case. 2George Green (1793-1841) was self educated and the son of a miller in Nottingham, England [184]. 344 Electromagnetic Field Theory 31.4.1 Scalar Waves Case Figure 31.6: The geometry for deriving Huygens’ principle for scalar wave equation. For a ψ(r) that satisfies the scalar wave equation (∇2 + k2) ψ(r) = 0, (31.4.1) the corresponding scalar Green’s function g(r, r0) satisfies (∇2 + k2) g(r, r0) = −δ(r − r0). (31.4.2) Next, we multiply (31.4.1) by g(r, r0) and (31.4.2) by ψ(r). And then, we subtract the resultant equations and integrating over a volume V as shown in Figure 31.6. There are two cases to consider: when r0 is in V , or when r0 is outside V . Thus, we have ¢ if r0 ∈ V , ψ(r0) = dr [g(r, r0)∇2ψ(r) − ψ(r)∇2g(r, r0)], (31.4.3) if r0 6∈ V , 0 V The left-hand side evaluates to different values depending on where r0 is due to the sifting property of the delta function δ(r−r0). Since g∇2ψ −ψ∇2g = ∇·(g∇ψ −ψ∇g), the left-hand side of (31.4.3) can be rewritten using Gauss’ divergence theorem, giving3 if r0 ∈ V , ψ(r0) = dS nˆ · [g(r, r0)∇ψ(r) − ψ(r)∇g(r, r0)], (31.4.4) if r0 6∈ V , 0 S where S is the surface bounding V . The above is the Green’s theorem, or the mathematical expression that once ψ(r) andn ˆ · ∇ψ(r) are known on S, then ψ(r0) away from S could be 3The equivalence of the volume integral in (31.4.3) to the surface integral in (31.4.4) is also known as Green’s theorem [89]. Equivalence Theorems, Huygens’ Principle 345 found. This is similar to the expression of equivalence principle wheren ˆ · ∇ψ(r) and ψ(r) are equivalence sources on the surface S. The first term on the right-hand side radiates via the Green’s function g(r, r0). Since this is a monopole field, this source is also called a monolayer or single layer source. The second term radiates, on the other hand, via the normal derivative of the Green’s function, namelyn ˆ·∇g(r, r0). Since the derivative of a Green’s function yields a dipole field, the second term corresponds to sources that radiate like dipoles pointing normally to the surface S. These sources ara also called double layer (or dipole layer) sources. These terminologies are prevalent in acoustics. The above mathematical expression also embodies the extinction theorem that says if r0 is outside V , the left-hand side evaluates to zero. Figure 31.7: The geometry for deriving Huygens’ principle for scalar wave. The radiation from the source can be replaced by equivalence sources on the surface S, and the field outside S can be calculated using (31.4.4). Also, the field is zero inside S from (31.4.4). This is the extinction theorem. If the volume V is bounded by S and Sinf as shown in Figure 31.7, then the surface integral in (31.4.4) should include an integral over Sinf . But when Sinf → ∞, all fields look 0 4 like plane wave, and ∇ → −rjkˆ on Sinf . Furthermore, g(r−r ) ∼ O(1/r), when r → ∞, and ψ(r) ∼ O(1/r), when r → ∞, if ψ(r) is due to a source of finite extent. Then, the integral over Sinf in (31.4.4) vanishes, and (31.4.4) is valid for the case shown in Figure 31.7 as well but with the surface integral over surface S only. Here, the field outside S at r0 is expressible in terms of the field on S. This is similar to the inside-out equivalence principle we have discussed previously Section 31.1.1, albeit this is for scalar wave case. Notice that in deriving (31.4.4), g(r, r0) has only to satisfy (31.4.2) for both r and r0 in V but no boundary condition has yet been imposed on g(r, r0). Therefore, if we further require 4The symbol “O” means “of the order.” 346 Electromagnetic Field Theory that g(r, r0) = 0 for r ∈ S, then (31.4.4) becomes ψ(r0) = − dS ψ(r)n ˆ · ∇g(r, r0), r0 ∈ V. (31.4.5) S On the other hand, if require additionally that g(r, r0) satisfies (31.4.2) with the boundary conditionn ˆ · ∇g(r, r0) = 0 for r ∈ S, then (31.4.4) becomes ψ(r0) = dS g(r, r0)n ˆ · ∇ψ(r), r0 ∈ V. (31.4.6) S Equations (31.4.4), (31.4.5), and (31.4.6) are various forms of Huygens’ principle, or equiv- alence principle for scalar waves (acoustic waves) depending on the definition of g(r, r0). Equa- tions (31.4.5) and (31.4.6) stipulate that only ψ(r) orn ˆ · ∇ψ(r) need be known on the surface S in order to determine ψ(r0). The above are analogous to the PEC and PMC equivalence principle considered previously in Sections 31.2 and 31.3. (Note that in the above derivation, k2 could be a function of position as well.) 31.4.2 Electromagnetic Waves Case Figure 31.8: Derivation of the Huygens’ principle for the electromagnetic case. One only needs to know the surface fields on surface S in order to determine the field at r0 inside V . The derivation of Huygens’ principle and Green’s theorem for the electromagnetic case is more complicated than the scalar wave case. But fortunately, this problem is mathematically homomorphic to the scalar wave case. In dealing with the requisite vector algebra, we have to remember to cross the t’s and dot the i’s, to carry ourselves carefully through the laborious and complicated vector algebra! Equivalence Theorems, Huygens’ Principle 347 In a source-free homogeneous region, an electromagnetic wave satisfies the vector wave equation ∇ × ∇ × E(r) − k2 E(r) = 0. (31.4.7) The analogue of the scalar Green’s function for the scalar wave equation is the dyadic Green’s function for the electromagnetic wave case [1, 32, 185, 186]. Moreover, the dyadic Green’s function satisfies the equation5 ∇ × ∇ × G(r, r0) − k2 G(r, r0) = I δ(r − r0). (31.4.8) It can be shown by direct back substitution that the dyadic Green’s function in free space is [186] ∇∇ G(r, r0) = I + g(r − r0) (31.4.9) k2 The above allows us to derive the vector Green’s theorem [1, 32,185]. Then, after post-multiplying (31.4.7) by G(r, r0), pre-multiplying (31.4.8) by E(r), sub- tracting the resultant equations and integrating the difference over volume V , considering two cases as we did for the scalar wave case, we have ¢ if r0 ∈ V , E(r0) = dV E(r) · ∇ × ∇ × G(r, r0) if r0 6∈ V , 0 V −∇ × ∇ × E(r) · G(r, r0) . (31.4.10) Next, using the vector identity that6 −∇ · E(r) × ∇ × G(r, r0) + ∇ × E(r) × G(r, r0) = E(r) · ∇ × ∇ × G(r, r0) − ∇ × ∇ × E(r) · G(r, r0), (31.4.11) then the integrand of (31.4.10) can be written as a divergence. With the help of Gauss’ divergence theorem, it can be written as if r0 ∈ V , E(r0) = − dS nˆ · E(r) × ∇ × G(r, r0) + ∇ × E(r) × G(r, r0) if r0 6∈ V , 0 S = − dS −E(r) × nˆ · ∇ × G(r, r0) + iωµ nˆ × H(r) · G(r, r0) . (31.4.12) S 5A dyad is an outer product between two vectors, and it behaves like a tensor, except that a tensor is more general than a dyad. A purist will call the above a tensor Green’s function, as the above is not a dyad in its strictest definition. 6This identity can be established by using the identity ∇ · (A × B) = B · ∇ × A − A · ∇ × B. We will have to let (31.4.11) act on an aribitrary constant vector to convert the dyad into a vector before applying this identity. The equality of the volume integral in (31.4.10) to the surface integral in (31.4.12) is also known as the vector Green’s theorem [32, 185]. Earlier form of this theorem was known as Franz formula [187]. 348 Electromagnetic Field Theory The above is just the vector analogue of (31.4.4). We have used the cyclic relation of dot and cross products to rewrite the last expression. Sincen ˆ × E andn ˆ × H are associated with surface magnetic current Ms and surface electric current Js, respectively, the above can be thought of having these equivalence surface currents radiating via the dyadic Green’s function. Again, notice that (31.4.12) is derived via the use of (31.4.8), but no boundary condition has yet been imposed on G(r, r0) on S even though we have given a closed form solution for the free-space case. The above is similar to the outside-in equivalence theorem we have derived in Section 31.1.2 using a Gedanken experiment. Now we have a mathematical derivation of the same theorem. Now, if we require the additional boundary condition thatn ˆ ×G(r, r0) = 0 for r ∈ S. This corresponds to a point source radiating in the presence of a PEC surface. Then (31.4.12) becomes E(r0) = − dS nˆ × E(r) · ∇ × G(r, r0), r0 ∈ V (31.4.13) S for it could be shown thatn ˆ × H · G = H · nˆ × G implying that the second term in (31.4.12) is zero. On the other hand, if we require thatn ˆ × ∇ × G(r, r0) = 0 for r ∈ S, then (31.4.12) becomes E(r0) = −iωµ dS nˆ × H(r) · G(r, r0), r0 ∈ V (31.4.14) S Equations (31.4.13) and (31.4.14) state that E(r0) is determined if eithern ˆ×E(r) orn ˆ×H(r) is specified on S. This is in agreement with the uniqueness theorem. These are the mathematical expressions of the PEC and PMC equivalence problems we have considered in the previous sections, Sections 31.2 and 31.3. The dyadic Green’s functions in (31.4.13) and (31.4.14) are for a closed cavity since bound- ary conditions are imposed on S for them. But the dyadic Green’s function for an unbounded, homogeneous medium, given in (31.4.10) can be written as 1 G(r, r0) = [∇ × ∇ × I g(r − r0) − I δ(r − r0)], (31.4.15) k2 ∇ × G(r, r0) = ∇ × I g(r − r0). (31.4.16) Then, (31.4.12), for r0 ∈ V and r0 6= r, becomes 1 E(r0) = −∇0 × dS g(r − r0)n ˆ×E(r) + ∇0 ×∇0 × dS g(r − r0)n ˆ×H(r). (31.4.17) iω S S 0 The above can be applied to the geometry in Figure 31.7 where r is enclosed in S and Sinf . However, the integral over Sinf vanishes by virtue of the radiation condition as for (31.4.4). Then, (31.4.17) relates the field outside S at r0 in terms of only the equivalence surface currents Ms = E × nˆ and Js =n ˆ × H on S. This is similar to the inside-out problem in the equivalence theorem (see Section 31.1.1). It is also related to the fact that if the radiation condition is satisfied, then the field outside of the source region is uniquely satisfied. Hence, this is also related to the uniqueness theorem. Lecture 32 Shielding, Image Theory The physics of electromagnetic shielding and electromagnetic image theory (also called image theorem) go hand in hand. They work by the moving of charges around so as to cancel the impinging fields. By understanding simple cases of shielding and image theory, we can gain enough insight to solve some real-world problems. For instance, the art of shielding is very important in the field of electromagnetic compatibility (EMC) and electromagnetic interference (EMI). In the modern age where we have more electronic components working side by side in a very compact environment, EMC/EMI become an increasingly challenging issue. These problems have to be solved using heuristics with a high dosage of physical insight. 32.1 Shielding We can understand shielding by understanding how electric charges move around in a conduc- tive medium. They move around to shield out the electric field, or cancel the impinging field inside the conductor. There are two cases to consider: the static case and the dynamic case. The physical arguments needed to understand these two cases are quite different. Moreover, since there are no magnetic charges around, the shielding of magnetic field is quite different from the shielding of electric field, as shall be seen below. 32.1.1 A Note on Electrostatic Shielding We begin with the simple case of electrostatic shielding. For electrostatic problems, a con- ductive medium suffices to produce surface charges that shield out the electric field from the conductive medium. If the electric field is not zero, then since J = σE, the electric current inside the conductor will keep flowing. The current will produce charges on the surface of the conductor to cancel the imping field, until inside the conductive medium E = 0. In this case, electric current ceases to flow in the conductor. In other words, when the field reaches the quiescent state, the charges redistribute them- selves so as to shield out the electric field, and that the total internal electric field, E = 0 at 349 350 Electromagnetic Field Theory equilibrium. And from Faraday’s law that tangential E field is continuous, thenn ˆ × E = 0 on the conductor surface sincen ˆ × E = 0 inside the conductor. Figure 32.1 shows the static elec- tric field, in the quiescent state, between two conductors (even though they are not PECs), and the electric field has to be normal to the conductor surfaces. Moreover, since E = 0 inside the conductor, ∇Φ = 0 implying that the potential is a constant inside a conductor at equilibrium. Figure 32.1: The objects can just be conductors, and in the quiescent state (static state), the tangential electric field will be zero on their surfaces. Also, E = 0 inside the conductor, or ∇Φ = 0, or Φ is a constant. 32.1.2 Relaxation Time The time it takes for the charges to move around until they reach their quiescent distribution such that E(t) = 0 is called the relaxation time. It is very much similar to the RC time constant of an RC circuit consisting of a resistor in series with a capacitor (see Figure 32.2). It can be proven that this relaxation time is related to ε/σ, but the proof is beyond the scope of this course and it is worthwhile to note that this constant has the same unit as the RC time constant of an RC circuit where a charged capacitor relaxes as exp(−t/τ) where the relaxation time τ = RC. Note that when σ → ∞, the relaxation time is zero. In other words, in a perfect conductor or a superconductor, the charges reorient themselves instantaneously if the external field is time-varying so that E(t) = 0 always. Electrostatic shielding or low-frequency shielding is important at low frequencies. The Faraday cage or Faraday shield is an important application of such a shielding (see Figure 32.3) [188]. However, if the conductor charges are induced by an external electric field that is time varying, then the charges have to constantly redistribute/re-orient themselves to try to shield out the incident time-varying electric field. Currents have to be constantly flowing around the conductor. Then the electric field cannot be zero inside the conductors as shown in Figure 32.4. In other words, an object with finite conductivity cannot shield out completely a time-varying electric field. It can be shown that the depth of penetration of the field into Shielding, Image Theory 351 Figure 32.2: The relaxation (or disappearance of accumulated charges) in a conductive object is similar to the relaxation of charges from a charged capacitance in an RC circuit as shown. the conductive object is about a skin depth δ = p2/(ωµσ). Or the lower the frequency ω or the conductivity σ, the large the penetration depth. For a perfect electric conductor (PEC) , E = 0 inside with the following argument: Because if E 6= 0, then J = σE where σ → ∞. Let us assume an infinitesimally small time-varying electric field in the PEC to begin with. It will induce an infinitely large electric current, and hence an infinitely large time-varying magnetic field. An infinite time-varying magnetic field in turn yields an infinite electric field that will drive an electric current, and these fields and current will be infinitely large. This is an unstable chain of events if it is true. Moreover, it will generate infinite energy in the system, which is not physical. Hence, the only possibility for a stable solution is for the time-varying electromagnetic fields to be zero inside a PEC. Thus, for the PEC, the charges can re-orient themselves instantaneously on the surface when the inducing electric fields from outside are time varying. In other words, the relaxation time ε/σ is zero. As a consequence, the time-varying electric field E is always zero inside PEC, and therefore,n ˆ × E = 0 on the surface of the PEC, even for time-varying fields. 32.2 Image Theory The image theory here in electromagnetics is quite different from that in optics. As mentioned before, when the frequency of the fields is high, the waves associated with the fields can be described by rays. Therefore ray optics can be used to solve many high-frequency problems. We can use ray optics to understand how an image is generated in a mirror. But the image theory in electromagnetics is quite different from that in ray optics. Image theory can be used to derived closed form solutions to boundary value problems when the geometry is simple and has a lot of symmetry. These closed form solutions in turn 352 Electromagnetic Field Theory Figure 32.3: Faraday cage demonstration on volunteers in the Palais de la D´ecouverte in Paris (courtesy of Wikipedia). When the cage is grounded, charges will surge from the ground to the cage surface so as to make the field inside the cage zero, or the potential is constant. Therefore, a grounded Faraday cage effectively shields the external fields from entering the cage. offer physical insight into the problems. This theory or method is also discussed in many textbooks [1,43,54, 65,79, 181, 189]. 32.2.1 Electric Charges and Electric Dipoles Image theory for a flat conductor surface or a half-space is quite easy to derive. To see that, we can start with electro-static theory of putting a positive charge above a flat plane. As mentioned before, for electrostatics, the plane or half-space does not have to be a perfect conductor, but only a conductor (or a metal). From the previous Section 32.1.1, the tangential static electric field on the surface of the conductor has to be zero. By the principle of linear superposition, the tangential static electric field can be canceled by putting an image charge of opposite sign at the mirror location of the original charge. This is shown in Figure 32.5. Now we can mentally add the total field due to these two charges. When the total static electric field due to the original charge and image charge is sketched, it will look like that in Figure 32.6. It is seen that the static electric field satisfies the boundary condition thatn ˆ × E = 0 at the conductor interface due to symmetry. An electric dipole is made from a positive charge placed in close proximity to a negative Shielding, Image Theory 353 Figure 32.4: If the source that induces the charges on the conductor is time varying, the current in the conductor is always nonzero so that the charges can move around to respond to the external time-varying charges. The two figures above show the orientation of the charges for two snap-shot in time. In other words, a time-varying field can penetrate the conductor to approximately within a skin-depth δ = p2/(ωµσ). charge. Using that an electric charge reflects to an electric charge of opposite polarity above a conductor, one can easily see that a static horizontal electric dipole reflects to a static horizontal electric dipole of opposite polarity. By the same token, a static vertical electric dipole reflects to static vertical electric dipole of the same polarity as shown in Figure 32.7. If this electric dipole is a Hertzian dipole whose field is time-varying, then one needs a PEC half-space to shield out the electric field. Also, the image charges will follow the original dipole charges instantaneously. Then the image theory for static electric dipoles over a half- space still holds true if the dipoles now become Hertzian dipoles, but the surface will have to be a PEC surface so that the fields can be shielded out instantaneously. 32.2.2 Magnetic Charges and Magnetic Dipoles A static magnetic field can penetrate a conductive medium. This is apparent from our experience when we play with a bar magnet over a copper sheet: the magnetic field from the magnet can still be experienced by iron filings put on the other side of the copper sheet. However, this is not the case for a time-varying magnetic field. Inside a conductive medium, a time-varying magnetic field will produce a time-varying electric field, which in turn produces the conduction current via J = σE. This is termed eddy current, which by Lenz’s law, repels the magnetic field from the conductive medium.1 Now, consider a static magnetic field penetrating into a perfect electric conductor, an minute amount of time variation will produce an electric field, which in turn produces an infinitely large eddy current. So the stable state for a static magnetic field inside a PEC is for it to be expelled from the perfect electric conductor. This in fact is what we observe when a magnetic field is brought near a superconductor. Therefore, for the static magnetic field, where B = 0 inside the PEC, thenn ˆ · B = 0 on the PEC surface (see Figure 32.8). 1The repulsive force occurs by virtue of energy conservation. Since “work done” is needed to set the eddy current in motion in the conductor, or to impart kinetic energy to the electrons forming the eddy current, a repulsive force is felt in Lenz’s law so that work is done in pushing the magnetic field into the conductive medium. 354 Electromagnetic Field Theory Figure 32.5: The use of image theory to solve the BVP of a point charge on top of a conductor. The boundary condition is thatn ˆ × E = 0 on the conductor surface. By placing a negative charge with respect to the original charge, by the principle of linear superposition, both of them produce a total field with no tangential component at the interface. Now, assuming a magnetic monopole exists, it will reflect to itself on a PEC surface so thatn ˆ · B = 0 as shown in Figure 32.8. Therefore, a magnetic charge reflects to a charge of similar polarity on the PEC surface. By extrapolating this to magnetic dipoles, they will reflect themselves to the magnetic dipoles as shown in Figure 32.9. A horizontal magnetic dipole reflects to a horizontal magnetic dipole of the same polarity, and a vertical magnetic dipole reflects to a vertical magnetic dipole of opposite polarity. Hence, a dipolar bar magnet can be levitated by a superconductor when this magnet is placed closed to it. This is also known as the Meissner effect [190], which is shown in Figure 32.10. A time-varying magnetic dipole can be made from a electric current loop. Over a PEC, a time-varying magnetic dipole will reflect the same way as a static magnetic dipole as shown in Figure 32.9. 32.2.3 Perfect Magnetic Conductor (PMC) Surfaces Magnetic conductor does not come naturally in this world since there are no free-moving magnetic charges around. Magnetic monopoles are yet to be discovered. On a PMC surface, by duality,n ˆ × H = 0. At low frequency, it can be mimicked by a high µ material. One can see that for magnetostatics, at the interface of a high µ material and air, the magnetic flux is approximately normal to the surface, resembling a PMC surface. High µ materials are hard to find at higher frequencies. Sincen ˆ × H = 0 on such a surface, no electric current can flow on such a surface. Hence, a PMC can be mimicked by a surface where no surface electric current can flow. This has been achieved in microwave engineering with a mushroom surface as shown in Figure 32.11 [192]. The mushroom structure Shielding, Image Theory 355 Figure 32.6: By image theory, the total electric of the original problem and the equivalent problem when we add the total electric field due to the original charge and the image charge. consisting of a wire and an end-cap, can be thought of as forming an LC tank circuit. Close to the resonance frequency of this tank circuit, the surface of mushroom structures essentially ∼ becomes open circuits with no or little current flowing on the surface, or Js = 0. In other words,n ˆ × H =∼ 0. This resembles a PMC, because with no surface electric current on this surface, the tangential magnetic field is small, the hallmark of a good magnetic conductor, by the duality principle. Mathematically, a surface that is dual to the PEC surface is the perfect magnetic conductor (PMC) surface. The magnetic dipole is also dual to the electric dipole. Thus, over a PMC surface, these electric and magnetic dipoles will reflect differently as shown in Figure 32.12. One can go through Gedanken experiments and verify that the reflection rules are as shown in the figure. 32.2.4 Multiple Images For the geometry shown in Figure 32.13, one can start with electrostatic theory, and convince oneself thatn ˆ × E = 0 on the metal surface with the placement of charges as shown. For conducting media, the charges will relax to the quiescent distribution after the relaxation time. For PEC surfaces, one can extend these cases to time-varying dipoles because the charges in the PEC medium can re-orient instantaneously (i.e. with zero relaxation time) to shield out or expel the E and H fields. Again, one can repeat the above exercise for magnetic charges, magnetic dipoles, and PMC surfaces. 356 Electromagnetic Field Theory Figure 32.7: By image theory, on a conductor surface, a horizontal static dipole reflects to one of opposite polarity, while a static vertical dipole reflects to one of the same polarity. If the dipoles are time-varying, then a PEC will have a same reflection rule. Figure 32.8: On a PEC surface, the requisite boundary condition isn ˆ · B = 0. Hence, a magnetic monopole on top of a PEC surface will have magnetic field distributed as shown. By image theory, such a field distribution can be obtained by adding a magnetic monopole of the same polarity at its image point. 32.2.5 Some Special Cases—Spheres, Cylinders, and Dielectric In- terfaces One curious case is for a static charge placed near a conductive sphere (or cylinder) as shown in 2 Figure 32.14. A charge of +Q reflects to a charge of −QI inside the sphere. For electrostatics, the sphere (or cylinder) need only be a conductor. However, this cannot be generalized to electrodynamics or a time-varying problem, because of the retardation effect: A time-varying dipole or charge will be felt at different points asymmetrically on the surface of the sphere from the original and image charges. Exact cancelation of the tangential electric field on the surface of the sphere or cylinder cannot occur for time-varying field. When a static charge is placed over a dielectric interface, image theory can be used to find the closed form solution. This solution can be derived using Fourier transform technique which we shall learn later [35]. It can also be extended to multiple interfaces. But image theory cannot be used for the electrodynamic case due to the different speed of light in different media, giving rise to different retardation effects. 2This is worked out in p. 48 and p. 49, Ramo et al [31]. Shielding, Image Theory 357 Figure 32.9: Using the rule of how magnetic monopole reflects itself on a PEC surface, the reflection rules for magnetic dipoles can be ascertained. Magnetic dipoles are often denoted by double arrows. Figure 32.10: On a PEC (superconducting) surface, a vertical magnetic dipole (formed by a small permanent bar magnet here) reflects to one of opposite polarity. Hence, the magnetic dipoles repel each other displaying the Meissner effect. The magnet, because of the repulsive force from its image, levitates above the superconductor (courtesy of Wikipedia [191]). 358 Electromagnetic Field Theory Figure 32.11: A mushroom structure operates like an LC tank circuit. At the right res- onant frequency, the surface resembles an open-circuit surface where no current can flow. Hence, tangential magnetic field is zero resembling perfect magnetic conductor (courtesy of Sievenpiper [192]). Figure 32.12: Reflection rules for electric and magnetic dipoles over a PMC surface. Magnetic dipoles are denoted by double arrows. Shielding, Image Theory 359 Figure 32.13: Image theory for multiple images [31]. Figure 32.14: Image theory for a point charge near a cylinder or a sphere can be found in closed form [31]. 360 Electromagnetic Field Theory Figure 32.15: A static charge over a dielectric interface can be found in closed form using Fourier transform technique. The solution is beyond the scope of this course. Lecture 33 High Frequency Solutions, Gaussian Beams When the frequency is very high, the wavelength of electromagnetic wave becomes very short. In this limit, the solutions to Maxwell’s equations can be found approximately. These solutions offer a very different physical picture of electromagnetic waves, and they are often used in optics where the wavelength is short. So it was no surprise that for a while, optical fields were thought to satisfy a very different equations from those of electricity and magnetism. Thus it came as a surprise that when it was later revealed that in fact, optical fields satisfy the same Maxwell’s equations as the fields from electricity and magnetism! In this lecture, we shall seek approximate solutions to Maxwell’s equations or the wave equations when the frequency is high or the wavelength is short. High frequency approximate solutions are important in many real-world applications. This is possible when the wavelength is much smaller than the size of the structure. This can occur even in the microwave regime where the wavelength is not that small, but much smaller than the size of the structure. This is the case when microwave interacts with reflector antennas for instance. It is also the transition from waves regime to the optics regime in the solutions of Maxwell’s equations. Often times, the term “quasi-optical” is used to describe the solutions in this regime. In the high frequency regime, or when we are far away from a source much larger than the Rayleigh distance (see Section 27.2.1), the field emanating from a source resembles a spherical wave. Moreover, when the wavelength is much smaller than the radius of curvature of the wavefront, the spherical wave can be approximated by a local plane wave. Thus we can imagine rays to be emanating from a finite source forming the spherical wave. The spherical wave will ultimately be approximated by plane waves locally at the observation point. This will simplify the solutions in many instances. For instance, ray tracing can be used to track how these rays can propagate, bounce, or “richohet” in a complex environment. In fact, it is now done in a movie industry to give “realism” to simulate the nuances of how light ray will bounce around in a room, and reflecting off objects. 361 362 Electromagnetic Field Theory Figure 33.1: Ray-tracing technique can be used in the movie industry to produce realism in synthetic images (courtesy of Wikipedia). 33.1 Tangent Plane Approximations We have learnt that reflection and transmission of waves at a flat surface can be solved in closed form. The important point here is the physics of phase matching. Due to phase matching, we have the law of reflection, transmission and Snell’s law [58].1 When a surface is not flat anymore, there is no closed form solution. But when a surface is curved, an approximate solution can be found. This is obtained by using a local tangent-plane approximation when the radius of curvature is much larger than the wavelength. Hence, this is a good approximation when the frequency is high or the wavelength is short. It is similar in spirit that we can approximate a spherical wave by a local plane wave at the spherical wave front when the wavelength is short compared to the radius of curvature of the wavefront. When the wavelength is short, phase matching happens locally, and the law of reflection, transmission, and Snell’s law are satisfied approximately as shown in Figure 33.2. The tangent plane approximation is the basis for the geometrical optics (GO) approximation [32,194]. In GO, light waves are replaced by light rays. As mentioned before, a light ray is a part of a spherical wave where locally, it can be approximated by a plane wave. The reflection and transmission of these rays at an interface is then estimated using the local tangent plane approximation and local Fresnel reflection and transmission coefficients. This is also the basis for lens or ray optics from which lens technology is derived (see Figure 33.3). It is also the basis for ray tracing for high-frequency solutions [195, 196].2 Many real world problems do not have closed-form solutions, and have to be treated with approximate methods. In addition to geometrical approximations mentioned above, asymptotic methods are also used to find approximate solutions. Asymptotic methods imply finding a solution when there is a large parameter in the problem. In this case, it is usually the frequency. Such high-frequency approximate methods are discussed in [197–201]. 1This law is also known in the Islamic world in 984 [193]. 2Please note that the tangent plane approximation is invalid near a sharp corner or an edge. The solution has to be augmented by additional diffracted wave coming from the edge or the corner. High Frequency Solutions, Gaussian Beams 363 Figure 33.2: In the tangent plane approximation, the surface where reflection and refraction occur is assumed to be locally flat. Thus, phase-matching is approximately satisfied, and hence, the law of reflection, transmittion, and Snell’s law are satisfied locally. Figure 33.3: Tangent plane approximations can also be made at dielectric interfaces so that Fresnel reflection and transmission coeffients can be used to ascertain the interaction of light rays with a lens. Also, one can use ray tracing to understand the physics of an optical lens (courtesy of Wikipedia). 33.2 Fermat’s Principle Fermat’s principle (1600s) [58,202] says that a light ray follows the path that takes the shortest time between two points.3 Since time delay is related to the phase shift, and that a light ray can be locally approximated by a plane wave, this can be stated that a plane wave follows the path that has a minimal phase shift. This principle can be used to derive law of reflection, transmission, and refraction for light rays. It can be used as the guiding principle for ray tracing as well. 3This eventually give rise to the principle of least action. 364 Electromagnetic Field Theory Figure 33.4: In Fermat’s principle, a light ray, when propagating from point A to point C, takes the path of least delay in time in the time domain, and delay in phase in the frequency domain. Given two points A and C in two different half spaces as shown in Figure 33.4. Then the phase delay between the two points, per Figure 33.4, can be written as4 P = ki · ri + kt · rt (33.2.1) In the above, ki is parallel to ri, so is kt is parallel to rt. As this is the shortest path with minimum phase shift or time delay, according to Fermat’s principle, another other path will be longer. In other words, if B were to move slightly to another point, a longer path with more phase shift or time delay will ensue, or that B is the stationary point of the path length or phase shift. Specializing (33.2.1) to a 2D picture, then the phase shift as a function of xi is stationary. In this Figure 33.4, we have xi + xt = const. Therefore, taking the derivative of (33.2.1) or the phase change with respect to xi, assuming that ki and kt do not change as B is moved slightly,5 ∂P = 0 = kix − ktx (33.2.2) ∂xi The above yields the law of refraction that kix = ktx, which is just Snell’s law; it can also be obtained by phase matching as have been shown earlier. This law was also known in the Islamic world to Ibn Sahl in 984 [193]. 4In this course, for wavenumber, we use k and β interchangeably, where k is prevalent in optics and β used in microwaves. 5 One can show that as the separations between A, B, and C are large, and the change in xi is ∆xi, the changes in ki and kt are small. The change in phase shift mainly comes from the change in xi. Alternatively, q 2 2 q 2 2 we can write P = kiri +ktrt, and let ri = xi + zi , and rt = xt + zt , and take the derivative with respect to to xi, one would also get the same answer. High Frequency Solutions, Gaussian Beams 365 33.2.1 Generalized Snell’s Law Figure 33.5: A phase screen which is position dependent can be made using nano-fabrication and design with commercial software for solving Maxwell’s equations. In such a case, one can derive a generalized Snell’s law to describe the diffraction of a wave by such a surface (courtesy of Capasso’s group [203]). Metasurfaces are prevalent these days due to our ability for nano-fabrication and numerical simulation. One of them is shown in Figure 33.5. Such a metasurface can be thought of as a phase screen, providing additional phase shift for the light as it passes through it. Moreover, the added phase shift can be controlled to be a function of position due to advent in nano- fabrication technology and commercial software for numerical simulation. To model this phase screen, we can add an additional function Φ(x, y) to (33.2.1), namely that P = ki · ri + kt · rt − Φ(xi, yi) (33.2.3) Now applying Fermat’s principle that there should be minimal phase delay, and taking the derivative of the above with respect to xi, one gets ∂P ∂Φ(xi, yi) = kix − ktx − = 0 (33.2.4) ∂xi ∂xi The above yields that the generalized Snell’s law [203] that ∂Φ(xi, yi) kix − ktx = (33.2.5) ∂xi It implies that the transmitted light can be directed to other angles due to the additional phase screen. 6 6Such research is being pursued by V. Shalaev and A. Boltasseva’s group at Purdue U [204]. 366 Electromagnetic Field Theory 33.3 Gaussian Beam We have seen previously that in a source-free medium ∇2A + ω2µεA = 0 (33.3.1) ∇2Φ + ω2µεΦ = 0 (33.3.2) The above are four scalar equations; and the Lorenz gauge ∇ · A = −jωµεΦ (33.3.3) connects A and Φ. We can examine the solution of A such that −jβz A(r) = A0(r)e (33.3.4) where A (r) is a slowly varying function while e−jβz is rapidly varying in the z direction. 0 √ (Here, β = ω µ is the wavenumber.) This is primarily a quasi-plane wave propagating predominantly in the z-direction. We know this to be the case in the far field of a source, but let us assume that this form persists less than the far field, namely, in the Fresnel zone as well. Taking the x component of (33.3.4), we have7 −jβz Ax(r) = Ψ(r)e (33.3.5) where Ψ(r) = Ψ(x, y, z) is a slowly varying envelope function of x, y, and z, whereas e−jβz is a rapidly varying function of z when β is large or the frequency is high. 33.3.1 Derivation of the Paraxial/Parabolic Wave Equation Substituting (33.3.5) into (33.3.1), and taking the double z derivative first, we arrive at 2 2 ∂ −jβz ∂ ∂ 2 −jβz 2 Ψ(x, y, z) e = 2 Ψ(x, y, z) − 2jβ Ψ(x, y, z) − β Ψ(x, y, z) e (33.3.6) ∂z | {z } | {z } ∂z ∂z slow fast Consequently, after substituting the above into the x component of (33.3.1), making use of the definition of ∇2, we obtain an equation for Ψ(r), the slowly varying envelope as ∂2 ∂2 ∂ ∂2 Ψ + Ψ − 2jβ Ψ + Ψ = 0 (33.3.7) ∂x2 ∂y2 ∂z ∂z2 where the last term containing β2 on the right-hand side of (33.3.6) cancels with the term coming from ω2µεA of (33.3.1). So far, no approximation has been made in the above equation. Since β is linearly proportional to frequency ω, when β → ∞ , or in the high frequency limit, 2 ∂ ∂ 2jβ Ψ Ψ (33.3.8) ∂z ∂z2 7Also, the wave becomes a transverse wave in the far field, and keeping the transverse component suffices. High Frequency Solutions, Gaussian Beams 367 where we have assumed that Ψ is a slowly varying function of z such that βΨ ∂/∂zΨ. In other words, (33.3.7) can be approximated by ∂2Ψ ∂2Ψ ∂Ψ + − 2jβ ≈ 0 (33.3.9) ∂x2 ∂y2 ∂z The above is called the paraxial wave equation. It is also called the parabolic wave equation.8 ∼ It implies that the β vector of the wave is approximately parallel to the z axis, or βz = β to be much greater than βx and βy, and hence, the name. 33.3.2 Finding a Closed Form Solution A closed form solution to the paraxial wave equation can be obtained by a simple trick.9 It is known that 0 e−jβ|r−r | A (r) = (33.3.10) x 4π|r − r0| is the exact solution to 2 2 ∇ Ax + β Ax = 0 (33.3.11) as long as r 6= r0. One way to ensure that r 6= r0 always is to let r0 = −zjbˆ , a complex number. Then (33.3.10) is always a solution to (33.3.11) for all r, because |r − r0|= 6 0 always as r0 is complex. Then, we should next make a paraxial approximation to the solution (33.3.10) by assuming that x2 + y2 z2. By so doing, it follows that |r − r0| = px2 + y2 + (z + jb)2 x2 + y2 1/2 = (z + jb) 1 + (z + jb)2 x2 + y2 ≈ (z + jb) + + ..., |z + jb| → ∞ (33.3.12) 2(z + jb) where Taylor series has been used in approximating the last term. And then using the above approximation in (33.3.10) yield −jβ(z+jb) 2 2 e −jβ x +y A (r) ≈ e 2(z+jb) ≈ e−jβzΨ(r) (33.3.13) x 4π(z + jb) By comparing the above with (33.3.5), we can identify 2 2 jb −jβ x +y Ψ(x, y, z) =∼ A e 2(z+jb) (33.3.14) 0 z + jb 8The paraxial wave equation, the diffusion equation and the Schrodinger equation are all classified as parabolic equations in mathematical parlance [35, 49, 205, 206]. 9Introduced by Georges A. Deschamps of UIUC [207]. 368 Electromagnetic Field Theory where A0 is used to absorbed the constant to simplify the expression. By separating the exponential part into the real part and the imaginary part, viz., x2 + y2 x2 + y2 z b = − j (33.3.15) 2(z + jb) 2 z2 + b2 z2 + b2 and writing the prefactor in terms of amplitude and phase, viz., jb 1 j tan−1( z ) = e b (33.3.16) z + jb p1 + z2/b2 we then have 2 2 2 2 −1 z x +y x +y A0 j tan ( ) −jβ 2 2 z −bβ 2 2 Ψ(x, y, z) =∼ e b e 2(z +b ) e 2(z +b ) (33.3.17) p1 + z2/b2 The above can be rewritten as 2 2 2 2 A0 −jβ x +y − x +y jψ Ψ(x, y, z) =∼ e 2R e w2 e (33.3.18) p1 + z2/b2 where A0 is a new constant introduced to absorb undesirable constants arising out of the algebra, and 2b z2 z2 + b2 z w2 = 1 + ,R = , ψ = tan−1 (33.3.19) β b2 z b For a fixed z, the parameters w, R, and ψ are constants. It is seen that the beam is Gaussian tapered in the x and y directions. Hence, w is the beam waist which varies with z, and it is q 2b smallest when z = 0, or w = w0 = β . x2+y2 And the term exp(−jβ 2R ) resembles the phase front of a spherical wave where R is its radius of curvature. This can be appreciated by studying a spherical wave front e−jβR, and make a paraxial wave approximation, namely, x2 + y2 z2 to get 1/2 x2+y2 −jβR −jβ(x2+y2+z2)1/2 −jβz 1+ e = e = e z2 2 2 2 2 −jβz−jβ x +y −jβz−jβ x +y ≈ e 2z ≈ e 2R (33.3.20) In the last approximation, we assume that z ≈ R in the paraxial approximation. The phase ψ defined in (33.3.19) changes linearly with z for small z, and saturates to a constant for large z. Then the phase of the entire wave is due to the exp(−jβz) in (33.3.13). A cross section of the electric field due to a Gaussian beam is shown in Figure 33.6. High Frequency Solutions, Gaussian Beams 369 Figure 33.6: Electric field of a Gaussian beam in the x-z plane frozen in time. The wave moves to the right as time increases; here, b/λ = 10/6 (courtesy of Haus, Electromagnetic Noise and Quantum Optical Measurements [82]). The narrowest beam waist is given by p w0/λ = b/(λπ). 33.3.3 Other solutions In general, the paraxial wave equation in (33.3.9) is of the same form as the Schr¨odinger equation which is of utmost importance in quantum theory. In recent years, the solution of this equation has made use of spill-over knowledge and terms from quantum theory, such as spin angular momentum (SAM) or orbital angular momentum (OAM). But it is a partial differential equation which can be solved by the separation of variables just like the Helmholtz wave equation. Therefore, in general, it has solutions of the form10 1/2 2 −N/2 1 −(x2+y2)/w2 −j β (x2+y2) j(m+n+1)Ψ Ψ (x, y, z) = 2 e e 2R e nm πn!m! w √ √ ·Hn x 2/w Hm y 2/w (33.3.21) where Hn(ξ) is a Hermite polynomial of order n. The solutions can also be expressed in terms of Laguere polynomials , namely, 1/2 2 1 −j β ρ2 −ρ2/w2 +j(n+m+1)Ψ jlφ Ψ (x, y, z) = min(n, m)! e 2R − e e e nm πn!m! w √ ! 2ρ 2ρ2 (−1)min(n,m) Ln−m (33.3.22) w min(n,m) w2 k where Ln(ξ) is the associated Laguerre polynomial. These gaussian beams have rekindled recent excitement in the community because, in addition to carrying spin angular momentum as in a plane wave, they can carry orbital angular momentum due to the complex transverse field distribution of the beams.11 They 10See F. Pampaloni and J. Enderlein [208]. 11See D.L. Andrew, Structured Light and Its Applications and articles therein [209]. 370 Electromagnetic Field Theory harbor potential for optical communications as well as optical tweezers to manipulate trapped nano-particles. Figure 33.7 shows some examples of the cross section (xy plane) field plots for some of these beams. They are richly endowed with patterns implying that they can be used to encode information. These lights are also called structured lights [209]. Figure 33.7: Examples of structured light. It can be used in encoding more information in optical communications (courtesy of L. Allen and M. Padgett’s chapter in J.L. Andrew’s book on structured light [209]). Lecture 34 Scattering of Electromagnetic Field The scattering of electromagnetic field is an important and fascinating topic. It especially enriches our understanding of the interaction of light wave with matter. The wavelength of vis- ible light is several hundred nanometers, with atoms and molecules ranging from nano-meters onward, light-matter interaction is richly endowed with interesting physical phenomena! A source radiates a field, and ultimately, in the far field of the source, the field resembles a spherical wave which in turn resembles a plane wave. When a plane wave impinges on an object or a scatterer, the energy carried by the plane wave is deflected to other directions which is the process of scattering. In the optical regime, the scattered light allows us to see objects, as well as admire all hues and colors that are observed of objects. In microwave, the scatterers cause the loss of energy carried by a plane wave. A proper understanding of scat- tering theory allows us to understand many physical phenomena around us. We will begin by studying Rayleigh scattering, which is scattering by small objects compared to wavelength. With Rayleigh scattering of a simple sphere, we can understand why the sky is blue and the sunset is red! 34.1 Rayleigh Scattering Rayleigh scattering is a solution to the scattering of light by small particles. These particles are assumed to be much smaller than wavelength of light. The size of water molecule is about 0.25 nm, while the wavelength of blue light is about 500 nm. Then a simple solution can be found by using quasi-static analysis in the vicinity of the small particle. This simple scattering solution can be used to explain a number of physical phenomena in nature (see Figure 34.1). For instance, why the sky is blue, the sunset so magnificently beautiful, how birds and insects can navigate themselves without the help of a compass. By the same token, it can also be used to explain why the Vikings, as a seafaring people, could cross the Atlantic Ocean over to Iceland without the help of a magnetic compass as the Chinese did in ancient times. 371 372 Electromagnetic Field Theory Figure 34.1: The magnificent beauty of nature can be partly explained by Rayleigh scattering [210, 211]. When a ray of light impinges on an object, we model the incident light as a plane elec- tromagnetic wave (see Figure 34.2). Without loss of generality, we can assume that the electromagnetic wave is polarized in the z direction and propagating in the x direction. We assume the particle to be a small spherical particle with permittivity εs and radius a. Essen- tially, the particle sees a constant field as the plane wave impinges on it. In other words, the particle feels an quasi-electrostatic field in the incident field. The incident field polarizes the particle, making it radiate like a Hertzian dipole. This is the gist of a scattering process: an incident field induces current (in this case, polarization current) on the scatterer. With the induced current, the scatterer re-radiates (or scatters). Figure 34.2: Geometry for studying the Rayleigh scattering problem. Scattering of Electromagnetic Field 373 34.1.1 Scattering by a Small Spherical Particle The incident field polarizes the particle making it look like an electric dipole. Since the incident field is time harmonic, the small electric dipole will oscillate and radiate like a Hertzian dipole in the far field. First, we will look at the solution in the vicinity of the scatterer, namely, in the near field. Then we will motivate the form of the solution in the far field of the scatterer. (Solving a boundary value problem by looking at the solutions in two different physical regimes, and then matching the solutions together is known as asymptotic matching.) A Hertzian dipole can be approximated by a small current source so that J(r) =zIlδ ˆ (r) (34.1.1) Without loss of generality, we have assumed the Hertzian dipole to be at the origin. In the above, we can let the time-harmonic current I = dq/dt = jωq, then Il = jωql = jωp (34.1.2) where the dipole moment p = ql. The vector potential A due to a Hertzian dipole, after substituting (34.1.1), is ¦ 0 µ 0 J(r ) −jβ|r−r0| A(r) = dr 0 e 4π V |r − r | µIl jωµql =z ˆ e−jβr =z ˆ e−jβr (34.1.3) 4πr 4πr where we have made use of the sifting property of the delta function in (34.1.1) when it is substituted into the above integral. Near Field The above gives the vector potential A due to a Hertzian dipole. Since the dipole is infinites- imally small, the above solution is both valid in the near field as well as the far field. Since the dipole moment ql is induced by the incident field, we need to relate ql to the amplitude of the incident electric field. To this end, we need to convert the above vector potential field to the near electric field of a small dipole. From prior knowledge, we know that the electric field is given by E = −jωA − ∇Φ. From dimensional analysis, the scalar potential term dominates over the vector potential term in the near field of the scatterer. Hence, we need to derive, for the corresponding scalar potential, the approximate solution. The scalar potential Φ(r) is obtained from the Lorenz gauge (see (23.2.23)) that ∇ · A = −jωµεΦ. Therefore, −1 Il ∂ 1 Φ(r) = ∇ · A = − e−jβr (34.1.4) jωµε jωε4π ∂z r 374 Electromagnetic Field Theory When we are close to the dipole, by assuming that βr 1, we can use a quasi-static approx- imation about the potential.1 Then ∂ 1 ∂ 1 ∂r ∂ 1 z 1 e−jβr ≈ = = − (34.1.5) ∂z r ∂z r ∂z ∂r r r r2 or after using that z/r = cos θ, ql Φ(r) ≈ cos θ (34.1.6) 4πεr2 which is the static dipole potential because we are in the near field of the dipole. This dipole induced in the small particle is formed in response to the incident field and its dipole potential given by the previous expression. In other words, the incident field polarizes the small particle into a small dipole. The incident field can be approximated by a constant local static electric field, Einc =zE ˆ i (34.1.7) This is the field that will polarize the small particle. It can also be an electric field between two parallel plates. The corresponding electrostatic potential for the incident field is then Φinc = −zEi (34.1.8) so that Einc ≈ −∇Φinc =zE ˆ i, as ω → 0. The scattered dipole potential from the spherical particle in the vicinity of it is quasi-static and is given by a3 Φ = E cos θ (34.1.9) sca s r2 which is the potential due to a static dipole. The electrostatic boundary value problem (BVP) has been previously solved and2 εs − ε Es = Ei (34.1.10) εs + 2ε Using (34.1.10) in (34.1.9), we get 3 εs − ε a Φsca = Ei 2 cos θ (34.1.11) εs + 2ε r On comparing with (34.1.6), one can see that the dipole moment induced by the incident field is that εs − ε 3 p = ql = 4πε a Ei = αEi (34.1.12) εs + 2ε where α is the polarizability of the small particle. 1This is the same as ignoring retardation effect. 2It was one of the homework problems. See also Section 8.3.6. Scattering of Electromagnetic Field 375 Far Field Now that we have learnt that a small particle is polarized by the incident field, which can be treated as a constant E field in its vicinity. In other words, the incident field induces a small dipole moment on the small particle. If the incident field is time-harmonic, the the small dipole will be time-oscillating and it will radiate like a time-varying Hertzian dipole whose far field is quite different from its near field (see Section 25.2). In the far field of the Hertzian dipole, we can start with 1 E = −jωA − ∇Φ = −jωA − ∇∇ · A (34.1.13) jωµε In the above, Lorenz gauge (see (23.2.23)) has been used to relate Φ to the vector potential A. But when we are in the far field, A behaves like a spherical wave which in turn behaves like a local plane wave if one goes far enough. Therefore, ∇ → −jβ = −jβrˆ. Using this approximation in (34.1.13), we arrive at ββ E =∼ −jω A − · A = −jω(A − rˆrˆ · A) = −jω(θAˆ + φAˆ ) (34.1.14) β2 θ φ where we have usedr ˆ = β/β. This is similar to the far field result we have derived in Section 26.1.2. 34.1.2 Scattering Cross Section From (34.1.3), and making use of (34.1.2), we see that Aφ = 0 while jωµql A = − e−jβr sin θ (34.1.15) θ 4πr Consequently, using (34.1.12) for ql, we have in the far field that3 2 3 ∼ ω µql −jβr 2 εs − ε a −jβr Eθ = −jωAθ = − e sin θ = −ω µε Eie sin θ (34.1.16) 4πr εs + 2ε r Using local plane-wave approximation that r ε 1 H =∼ E = E (34.1.17) φ µ θ η θ where η = pµ/ε. The time-averaged Poynting vector is given by hSi = 1/2 But ¢ ¢ ¢ π π π sin3 θdθ = − sin2 θd cos θ = − (1 − cos2 θ)d cos θ 0 0 ¢0 −1 4 = − (1 − x2)dx = (34.1.20) 1 3 Therefore 2 4π εs − ε 4 6 2 Ps = β a |Ei| (34.1.21) 3η εs + 2εs In the above, even though we have derived the equation using electrostatic theory, it is also valid for complex permittivity defined in Section 7.1.2. One can take the divergence of (7.1.9) to arrive at a Gauss’ law for lossy dispersive media, viz., ∇ · εE = 0 which is homomorphic to the lossless case. Hurrah again to phasor technique! e The scattering cross section is the effective area of a scatterer such that the total scattered power is proportional to the incident power density times the scattering cross section. As such it is defined as 2 2 Ps 8πa εs − ε 4 Σs = = (βa) (34.1.22) hSinci 3 εs + 2ε where we have used the local plane-wave approximation that 1 hS i = |E |2 (34.1.23) inc 2η i The above also implies that Ps = hSinci · Σs In other words, the scattering cross section Σs is an effective cross-sectional area of the scatterer that will intercept the incident wave power hSinci to produce the scattered power Ps. It is seen that the scattering cross section grows as the fourth power of frequency since β = ω/c. The radiated field grows as the second power because it is proportional to the acceleration of the charges on the particle. The higher the frequency, the more the scattered power. This mechanism can be used to explain why the sky is blue. It also can be used to explain why sunset has a brilliant hue of red and orange (see Figure 34.3). Scattering of Electromagnetic Field 377 Figure 34.3: During the day time, when we look at the sky, we mainly see scattered sunlight. Since high-frequency light is scattered more, the sky appears blue. At sunset, the sunlight has to go through a thicker atmosphere. Thus, the blue light is scattered away, leaving the red light that reaches the eyes. Hence, the sunset appears red. The above also explains the brilliant glitter of gold plasmonic nano-particles as discovered by ancient Roman artisans. For gold, the medium resembles a plasma, and hence, we can have εs < 0, and the denominator can be very small giving rise to strongly scattered light (see Section 8.3.6). Furthermore, since the far field scattered power density of this particle is 1 hSi = E H∗ ∼ sin2 θ (34.1.24) 2η θ φ the scattering pattern of this small particle is not isotropic. In other words, these dipoles ra- diate predominantly in the broadside direction but not in their end-fire directions. Therefore, insects and sailors can use this to figure out where the sun is even in a cloudy day. In fact, it is like a rainbow: If the sun is rising or setting in the horizon, there will be a bow across the sky where the scattered field is predominantly linearly polarized.4 Such a “sunstone” for direction finding is shown in Figure 34.4. 4You can go through a Gedanken experiment to convince yourself of such. 378 Electromagnetic Field Theory Figure 34.4: A sunstone can indicate the polarization of the scattered light. From that, one can deduce where the sun is located (courtesy of Wikipedia). 34.1.3 Small Conductive Particle The above analysis is for a small dielectric particle. The quasi-static analysis may not be valid for when the conductivity of the particle becomes very large. For instance, for a perfect electric conductor immersed in a time varying electromagnetic field, the magnetic field in the long wavelength limit induces eddy current in PEC sphere.5 Hence, in addition to an electric dipole component, a PEC sphere also has a magnetic dipole component. The scattered field due to a tiny PEC sphere is a linear superposition of an electric and magnetic dipole components. These two dipolar components have electric fields that cancel precisely at certain observation angle. It gives rise to deep null in the bi-static radar scattering cross-section (RCS)6 of a PEC sphere as illustrated in Figure 34.5. 5Note that there is no PEC at optics. A metal behaves more like a plasma medium at optical frequencies. 6Scattering cross section in microwave range is called an RCS due to its prevalent use in radar technology. Scattering of Electromagnetic Field 379 Figure 34.5: RCS (radar scattering cross section) of a small PEC scatterer (courtesy of Sheng et al. [212]). 34.2 Mie Scattering When the size of the dipole becomes larger compared to wavelength λ, quasi-static approxi- mation is insufficient to approximate the solution. Then one has to solve the boundary value problem in its full glory usually called the full-wave theory or Mie theory [213,214]. With this theory, the scattering cross section does not grow indefinitely with frequency as in (34.1.22). It has to saturate to a value for increasing frequency. For a sphere of radius a, the scattering cross section becomes πa2 in the high-frequency limit. This physical feature of this plot is shown in Figure 34.6, and it also explains why the sky is not purple. 380 Electromagnetic Field Theory Figure 34.6: Radar cross section (RCS) calculated using Mie scattering theory [214]. 34.2.1 Optical Theorem Before we discuss the Mie scattering solution, let us discuss an amazing theorem called the optical theorem. This theorem says that the scattering cross section of a scatterer depends only on the forward scattering power density of the scatterer. In other words, if a plane wave is incident on a scatterer, the scatterer will scatter the incident power in all directions. But the total power scattered by the object is only dependent on the forward scattering power density of the object or scatterer. This amazing theorem is called the optical theorem, and the proof of this is given in J.D. Jackson’s book [48]. The true physical reason for this is power orthogonality. Two plane waves cannot interact or exchange power with each other unless they share the same k or β vector, where β is both the plane wave direction of the incident wave as well as the forward scattered wave. This is similar to power orthogonality in a waveguide, and it happens for orthogonal modes in waveguides [83, 178]. The scattering pattern of a scatterer for increasing frequency is shown in Figure 34.7. For Rayleigh scattering where the wavelength is long, the scattered power is distributed isotrop- ically save for the doughnut shape of the radiation pattern, namely, the sin2(θ) dependence. As the frequency increases, the power is scattered increasingly in the forward direction. The reason being that for very short wavelength, the scatterer looks like a disc to the incident Scattering of Electromagnetic Field 381 wave, casting a shadow in the forward direction. Hence, there has to be scattered field in the forward direction to cancel the incident wave to cast this shadow. In a nutshell, the scattering theorem is intuitively obvious for high-frequency scattering. The amazing part about this theorem is that it is true for all frequencies. Figure 34.7: A particle scatters increasingly more in the forward direction as the frequency increases (Courtesy of hyperphysics.phy-astr.gsu.edu). 34.2.2 Mie Scattering by Spherical Harmonic Expansions As mentioned before, as the wavelength becomes shorter, we need to solve the boundary value problem in its full glory without making any approximations. This closed form solution can be found for a sphere scattering by using separation of variables and spherical harmonic expansions that will be discussed in the section. The Mie scattering solution by a sphere is beyond the scope of this course.7 The separation of variables in spherical coordinates is not the only useful for Mie scattering, it is also useful for analyzing spherical cavity. So we will present the precursor knowledge so that you can read further into Mie scattering theory if you need to in the future. 34.2.3 Separation of Variables in Spherical Coordinates8 To this end, we look at the scalar wave equation (∇2 +β2)Ψ(r) = 0 in spherical coordinates. A lookup table can be used to evaluate ∇·∇, or divergence of a gradient in spherical coordinates. Hence, the Helmholtz wave equation becomes9 1 ∂ ∂ 1 ∂ ∂ 1 ∂2 r2 + sin θ + + β2 Ψ(r) = 0 (34.2.1) r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2 Noting the ∂2/∂φ2 derivative, by using separation of variables technique, we assume Ψ(r) to be of the form Ψ(r) = F (r, θ)ejmφ (34.2.2) 7But it is treated in J.A. Kong’s book [32] and Chapter 3 of W.C. Chew, Waves and Fields in Inhomogeneous Media [35] and many other textbooks [48, 65, 181]. 8May be skipped on first reading. 9By quirk of mathematics, it turns out that the first term on the right-hand side below can be simplified by observing that 1 ∂ r2 = 1 ∂ r. r2 ∂r r ∂r 382 Electromagnetic Field Theory ∂2 jmφ This will simplify the ∂/∂φ derivative in the partial differential equation since ∂φ2 e = −m2ejmφ. Then (34.2.1) becomes 1 ∂ ∂ 1 ∂ ∂ m2 r2 + sin θ − + β2 F (r, θ) = 0 (34.2.3) r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ Again, by using the separation of variables, and letting further that m F (r, θ) = bn(βr)Pn (cos θ) (34.2.4) where we require that 1 d d m2 sin θ + n(n + 1) − P m(cos θ) = 0 (34.2.5) sin θ dθ dθ sin2 θ n m when Pn (cos θ) is the associate Legendre polynomial. Note that (34.2.5) is an eigenvalue problem with eigenvalue n(n + 1), and |m| ≤ |n|. The value n(n + 1) is also known as separation constant. Consequently, bn(kr) in (34.2.4) satisfies 1 d d n(n + 1) r2 − + β2 b (βr) = 0 (34.2.6) r2 dr dr r2 n The above is the spherical Bessel equation where bn(βr) is either the spherical Bessel function (1) jn(βr), spherical Neumann function nn(βr), or the spherical Hankel functions, hn (βr) and (2) hn (βr). The spherical functions are the close cousins of the cylindrical functions. They are related to the cylindrical functions via [35,49] 10 r π bn(βr) = Bn+ 1 (βr) (34.2.7) 2βr 2 It is customary to define the spherical harmonic as [48,110] s 2n + 1 (n − m)! Y (θ, φ) = P m(cos θ)ejmφ (34.2.8) nm 4π (n + m)! n The above is normalized such that m ∗ Yn,−m(θ, φ) = (−1) Ynm(θ, φ) (34.2.9) and that ¢ ¢ 2π π ∗ dφ sin θdθYn0m0 (θ, φ)Ynm(θ, φ) = δn0nδm0m (34.2.10) 0 0 These functions are also complete11 like Fourier series, so that ∞ n X X ∗ 0 0 0 0 Ynm(θ , φ )Ynm(θ, φ) = δ(φ − φ )δ(cos θ − cos θ ) (34.2.11) n=0 m=−n 10By a quirk of nature, the spherical Bessel functions needed for 3D wave equations are in fact simpler than cylindrical Bessel functions needed for 2D wave equation. One can say that 3D is real, but 2D is surreal. 11In a nutshell, a set of basis functions is complete in a subspace if any function in the same subspace can be expanded as a sum of these basis functions. Lecture 35 Spectral Expansions of Source Fields—Sommerfeld Integrals In previous lectures, we have assumed plane waves in finding closed form solutions. Plane waves are simple waves, and their reflections off a flat surface or a planarly layered medium can be found easily. When we have a source like a point source, it generates a spherical wave. We do not know how to reflect exactly a spherical wave off a planar interface. But by expanding a spherical wave in terms of sum of plane waves and evanescennt waves using Fourier transform technique, we can solve for the solution of a point source over a layered medium easily in terms of spectral integrals. Sommerfeld was the first person to have done this, and hence, these integrals are often called Sommerfeld integrals. Finally, we shall apply the method of stationary phase to approximate these integrals to elucidate their physics. From this, we can see ray theory emerging from the complicated mathematics. It reminds me of a lyric from the musical The Sound of Music—Ray, a drop of golden sun! Ray has mesmerized the human mind, and it will be interesting to see if the mathematics behind it is equally enchanting. By this time, you probably feel inundated by the ocean of knowledge that you are imbibing. If you can assimilate them, it will be an exhilarating experience. 35.1 Spectral Representations of Sources A plane wave is a mathematical idealization that does not exist in the real world. In practice, waves are nonplanar in nature as they are generated by finite sources, such as antennas and scatterers: For example, a point source generates a spherical wave which is nonplanar. Fortunately, these waves can be expanded in terms of sum of plane waves. Once this is done, then the study of non-plane-wave reflections from a layered medium becomes routine. In the following, we shall show how waves resulting from a point source can be expanded in terms of plane waves summation. This topic is found in many textbooks [1,32,35,95,96,181,205,215]. 383 384 Electromagnetic Field Theory 35.1.1 A Point Source There are a number of ways to derive the plane wave expansion of a point source. We will illustrate one of the ways. The spectral decomposition or the plane-wave expansion of the field due to a point source could be derived using Fourier transform technique. First, notice that the scalar wave equation with a point source at the origin is ∂2 ∂2 ∂2 ∇2 + k2 φ(x, y, z) = + + + k2 φ(x, y, z) = −δ(x) δ(y) δ(z). (35.1.1) 0 ∂x2 ∂y2 ∂z2 0 The above equation could then be solved in the spherical coordinates, yielding the solution given in the previous lecture, namely, Green’s function with the source point at the origin, or1 eik0r φ(x, y, z) = φ(r) = . (35.1.2) 4πr The solution is entirely spherically symmetric due to the symmetry of the point source. Next, assuming that the Fourier transform of φ(x, y, z) exists,2 we can write ¦∞ 1 φ(x, y, z) = dk dk dk φ˜(k , k , k )eikxx+iky y+ikz z. (35.1.3) (2π)3 x y z x y z −∞ Then we substitute the above into (35.1.1), after exchanging the order of differentiation and integration,3 one can simplify the Laplacian operator in the Fourier space, or spectral domain, to arrive at ∂2 ∂2 ∂2 ∇2 = + + = −k2 − k2 − k2 ∂x2 ∂y2 ∂z2 x y z Then, together with the Fourier representation of the delta function, which is4 ¦∞ 1 δ(x) δ(y) δ(z) = dk dk dk eikxx+iky y+ikz z (35.1.4) (2π)3 x y z −∞ we convert (35.1.1) into ¦∞ 2 2 2 2 ˜ ikxx+iky y+ikz z dkxdkydkz [k0 − kx − ky − kz ]φ(kx, ky, kz)e (35.1.5) −∞ ¦∞ ikxx+iky y+ikz z = − dkxdkydkz e . (35.1.6) −∞ 1From this point onward, we will adopt the exp(−iωt) time convention to be commensurate with the optics and physics literatures. ¡ 2 ∞ The Fourier transform of a function f(x) exists if it is absolutely integrable, namely that −∞ |f(x)|dx is finite (see [110]). 3Exchanging the order of differentiation and integration is allowed if the integral converges after the exchange. ¡ 4 ∞ We have made use of that δ(x) = 1/(2π) −∞ dkx exp(ikxx) three times. Spectral Expansions of Source Fields—Sommerfeld Integrals 385 Since the above is equal for all x, y, and z, we can Fourier inverse transform the above to get ˜ −1 φ(kx, ky, kz) = 2 2 2 2 . (35.1.7) k0 − kx − ky − kz Consequently, we have ¦∞ −1 eikxx+iky y+ikz z φ(x, y, z) = 3 dk 2 2 2 2 . (35.1.8) (2π) k0 − kx − ky − kz −∞ where dk = dkxdkydkz. The above expresses the fact the φ(x, y, z) which is a spherical wave by (35.1.2), is expressed as an integral summation of plane waves. But these plane waves are 2 2 2 2 not physical plane waves in free space since kx + ky + kz 6= k0. Im [kz] C k2 – k 2 – k 2 ⊗ 0 x y × Re [kz] 2 2 2 Fourier Inversion Contour – k0 – k x – k y Figure 35.1: The integration along the real axis is equal to the integration along C plus the 2 2 2 1/2 residue of the pole at (k0 − kx − ky) , by invoking Jordan’s lemma. Weyl Identity To make the plane waves in (35.1.8) into physical plane waves, we have to massage it into a different form. We rearrage the integrals in (35.1.8) so that the dkz integral is performed first. In other words, ¤∞ ¢ 1 ∞ eikz z ikxx+iky y φ(r) = 3 dkxdkye dkz 2 2 2 2 (35.1.9) (2π) −∞ kz − (k0 − kx − ky) −∞ where we have deliberately rearrange the denominator with kz being the variable in the inner 2 2 2 1/2 5 integral. Then the integrand has poles at kz = ±(k0 − kx − ky) . Moreover, for real k0, and real values of kx and ky, these two poles lie on the real axis, rendering the integral in 5 2 2 2 2 In (35.1.8), the pole is located at kx + ky + kz = k0. This equation describes a sphere in k space, known as the Ewald’s sphere [216]. 386 Electromagnetic Field Theory 0 00 (35.1.8) undefined. However, if a small loss is assumed in k0 such that k0 = k0 + ik0 , then the poles are off the real axis (see Figure 35.1), and the integrals in (35.1.8) are well-defined. In actual fact, this is intimately related to the uniqueness principle we have studied before: An infinitesimal loss is needed to guarantee uniqueness in an open space as shall be explained below. First, the reason is that without loss, |φ(r)| ∼ O(1/r), r → ∞ is not strictly absolutely integrable, and hence, its Fourier transform does not exist [52]: The manipulation that leads to (35.1.8) is not strictly correct. Second, the introduction of a small loss also guarantees the radiation condition and the uniqueness of the solution to (35.1.1), and therefore, the equality of (35.1.2) and (35.1.8) [35]. Observe that in (35.1.8), when z > 0, the integrand is exponentially small when =m[kz] → ∞. Therefore, by Jordan’s lemma [89], the integration for kz over the contour C as shown in Figure 35.1 vanishes. Then, by Cauchy’s theorem [89], the integration over the Fourier inversion contour on the real axis is the same as integrating over the pole singularity located 2 2 2 1/2 at (k0 − kx − ky) , yielding the residue of the pole (see Figure 35.1). Consequently, after doing the residue evaluation, we have ¤∞ ik x+ik y+ik0 z i e x y z φ(x, y, z) = 2 dkxdky 0 , z > 0, (35.1.10) 2(2π) kz −∞ 0 2 2 2 1/2 where kz = (k0 − kx − ky) is the value of kz at the pole location. Similarly, for z < 0, we can add a contour C in the lower-half plane that contributes zero to the integral, one can deform the contour to pick up the pole contribution. Hence, the 0 2 2 2 1/2 integral is equal to the pole contribution at kz = −(k0 − kx − ky) (see Figure 35.1). As such, the result for all z can be written as ¤∞ ik x+ik y+ik0 |z| i e x y z φ(x, y, z) = 2 dkxdky 0 , all z. (35.1.11) 2(2π) kz −∞ By the uniqueness of the solution to the partial differential equation (35.1.1) satisfying radiation condition at infinity, we can equate (35.1.2) and (35.1.11), yielding the identity ¤∞ eik0r i eikxx+iky y+ikz |z| = dkxdky , (35.1.12) r 2π kz −∞ 2 2 2 2 2 2 2 1/2 where kx +ky +kz = k0, or kz = (k0 −kx −ky) . The above is known as the Weyl identity (Weyl 1919). To ensure the radiation condition, we require that =m[kz] > 0 and Figure 35.2: The wave is propagating for kρ vectors inside the disk, while the wave is evanes- cent for kρ outside the disk. One can also interpret the above as a 2D surface integral in the Fourier space over the 2 2 2 kx and ky plane or variables. When kx + ky < k0, or the spatial spectrum is inside a disk of radius k0, the waves are propagating waves. But for contributions outside this disk, the waves are evanescent (see Figure 35.2). And the high Fourier (or spectral) components of the Fourier spectrum correspond to evanescent waves. The high spectral components, which are related to the evanescent waves, are important for reconstructing the singularity of the Green’s function. y kρ a ρ φ x Figure 35.3: The kρ and the ρ vector on the xy plane. Sommerfeld Identity The Weyl identity has double integral, and hence, is more difficult to integrate numerically. Here, we shall derive the Sommerfeld identity which has only one integral. First, in (35.1.12), we express the integral in cylindrical coordinates and write kρ =xk ˆ ρ cos α +yk ˆ ρ sin α, ρ = xρˆ cos φ +yρ ˆ sin φ (see Figure 35.3), and dkxdky = kρdkρ dα. Then, kxx + kyy = kρ · ρ = 388 Electromagnetic Field Theory kρ cos(α − φ), and with the appropriate change of variables, we have ¢ ∞ ¢ eik0r i 2π eikρρ cos(α−φ)+ikz |z| = kρdkρ dα , (35.1.13) r 2π 0 kz 0 2 2 2 1/2 2 2 1/2 where kz = (k0 −kx −ky) = (k0 −kρ) , where in cylindrical coordinates, in the kρ-space, 2 2 2 or the Fourier space, kρ = kx + ky. Then, using the integral identity for Bessel functions given by6 ¢2π 1 J (k ρ) = dα eikρρ cos(α−φ), (35.1.14) 0 ρ 2π 0 (35.1.13) becomes ¢ ∞ eik0r k ρ ikz |z| = i dkρ J0(kρρ)e . (35.1.15) r kz 0 The above is also known as the Sommerfeld identity (Sommerfeld 1909 [109]; [205][p. 242]). Its physical interpretation is that a spherical wave can now be expanded as an integral summation of conical waves or cylindrical waves in the ρ direction, times a plane wave in the z direction over all wave numbers kρ. This wave is evanescent in the ±z direction when kρ > k0. (1) (2) By using the fact that J0(kρρ) = 1/2[H0 (kρρ) + H0 (kρρ)], and the reflection formula (1) iπ (2) that H0 (e x) = −H0 (x), a variation of the above identity can be derived as ¢ ∞ eik0r i k ρ (1) ikz |z| = dkρ H0 (kρρ)e . (35.1.16) r 2 kz −∞ Im [kρ] • +k0 Re [k ] • ρ –k 0 Sommerfeld Integration Path Figure 35.4: Sommerfeld integration path. (1) 7 2 2 1/2 Since H0 (x) has a logarithmic branch-point singularity at x = 0, and kz = (k0 − kρ) has algebraic branch-point singularities at kρ = ±k0, the integral in Equation (35.1.16) is 6See Chew [35], or Whitaker and Watson(1927) [217]. 7 (1) 2i H0 (x) ∼ π ln(x), see Chew [35][p. 14], or Abromawitz or Stegun [117]. Spectral Expansions of Source Fields—Sommerfeld Integrals 389 undefined unless we stipulate also the path of integration. Hence, a path of integration adopted by Sommerfeld, which is even good for a lossless medium, is shown in Figure 35.4. Because of the manner in which we have selected the reflection formula for Hankel functions, (1) iπ (2) i.e., H0 (e x) = −H0 (x), the path of integration should be above the logarithmic branch- point singularity at the origin. With this definition of the Sommerfeld integration, the integral is well defined even when there is no loss, i.e., when the bronach points ±k0 are on the real axis. 35.2 A Source on Top of a Layered Medium Previously, we have studied the propagation of plane electromagnetic waves from a single dielectric interface in Section 14.1 as well as through a layered medium in Section 16.1. It can be shown that plane waves reflecting from a layered medium can be decomposed into TE-type plane waves, where Ez = 0, Hz 6= 0, and TM-type plane waves, where Hz = 0, 8 Ez 6= 0. One also sees how the field due to a point source can be expanded into plane waves in Section 35.1. In view of the above observations, when a point source is on top of a layered medium, it is then best to decompose its field in terms of plane waves of TE-type and TM-type. Then, the nonzero component of Ez characterizes TM waves, while the nonzero component of Hz characterizes TE waves. Hence, given a field, its TM and TE components can be extracted readily. Furthermore, if these TM and TE components are expanded in terms of plane waves, their propagations in a layered medium can be studied easily. The problem of a vertical electric dipole on top of a half space was first solved by Som- merfeld (1909) [109] using Hertzian potentials, which are related to the z components of the electromagnetic field. The work is later generalized to layered media, as discussed in the liter- ature. Later, Kong (1972) [218] suggested the use of the z components of the electromagnetic field instead of the Hertzian potentials. 35.2.1 Electric Dipole Fields–Spectral Expansion The representation of a spherical wave in terms of plane waves can be done using Weyl identity or Sommerfeld identiy. Here, we will use Sommerfeld identity in anticipation of simpler numerical integration, since only single integrals are involved. The E field in a homogeneous medium due to a point current source or a Hertzian dipole directed in theα ˆ direction, J =αI` ˆ δ(r), is derivable via the vector potential method or the dyadic Green’s function approach. Then, using the dyadic Green’s function approach, or the vector/scalar potential approach, the field due to a Hertzian dipole is given by ∇∇ eikr E(r) = iωµ I + · αI`ˆ , (35.2.1) k2 4πr √ where I` is the current moment and k = ω µ , the wave number of the homogeneous medium. Furthermore, from ∇ × E = iωµH, the magnetic field due to a Hertzian dipole is 8Chew, Waves and Fields in Inhomogeneous Media [35]; Kong, Electromagnetic Wave Theory [32]. 390 Electromagnetic Field Theory shown to be given by eikr H(r) = ∇ × αI`ˆ . (35.2.2) 4πr With the above fields, their TM and TE components can be extracted easily in anticipation of their plane wave expansions for propagation through layered media. (a) Vertical Electric Dipole (VED) z Region 1 x –d1 Region i –di Figure 35.5: A vertical electric dipole over a layered medium. A vertical electric dipole shown in Figure 35.5 hasα ˆ =z ˆ; hence, in anticipation of their plane wave expansions, the TM component of the field is characterized by Ez 6= 0 or that iωµI` ∂2 eikr E = k2 + , (35.2.3) z 4πk2 ∂z2 r and the TE component of the field is characterized by Hz = 0, (35.2.4) implying the absence of the TE field. Next, using the Sommerfeld identity (35.1.16) in the above, and after exchanging the order of integration and differentiation, we have9 ¢∞ −I` k3 ρ ikz |z| Ez = dkρ J0(kρρ)e , |z|= 6 0 (35.2.5) 4πω kz 0 2 2 2 after noting that kρ + kz = k . Notice that now Equation (35.2.5) expands the z component of the electric field in terms of cylindrical waves in the ρ direction and a plane wave in the z direction. Since cylindrical waves actually are linear superpositions of plane waves, because we can work backward from (35.1.16) to (35.1.12) to see this. As such, the integrand in 9By using (35.1.16) in (35.2.3), the ∂2/∂z2 operating on eikz |z| produces a Dirac delta function singularity. Detail discussion on this can be found in the chapter on dyadic Green’s function in Chew, Waves and Fields in Inhomogeneous Media [35]. Spectral Expansions of Source Fields—Sommerfeld Integrals 391 (35.2.5) in fact consists of a linear superposition of TM-type plane waves. The above is also the primary field generated by the source.10 Consequently, for a VED on top of a stratified medium as shown, the downgoing plane wave from the point source will be reflected like TM waves with the generalized reflection ˜TM coefficient R12 . Hence, over a stratified medium, the field in region 1 can be written as ¢ ∞ 3 −I` kρ h i ik1z |z| ˜TM ik1z z+2ik1z d1 E1z = dkρ J0(kρρ) e + R12 e , (35.2.6) 4πω1 k1z 0 2 2 1 2 2 where k1z = (k1 − kρ) 2 , and k1 = ω µ11, the wave number in region 1. The phase-matching condition dictates that the transverse variation of the field in all the regions must be the same. Consequently, in the i-th region, the solution becomes ¢ ∞ 3 −I` kρ h i −ikiz z ˜TM ikiz z+2ikiz di iEiz = dkρ J0(kρρ)Ai e + Ri,i+1e . (35.2.7) 4πω k1z 0 Notice that Equation (35.2.7) is now expressed in terms of iEiz because iEiz reflects and 11 transmits like Hiy, the transverse component of the magnetic field or TM waves. Therefore, ˜TM Ri,i+1 and Ai could be obtained using the methods discussed in Chew, Waves and Fields in Inhomogeneous Media [110]. This completes the derivation of the integral representation of the electric field everywhere in the stratified medium. These integrals are known as Sommerfeld integrals. The case when the source is embedded in a layered medium can be derived similarly. (b) Horizontal Electric Dipole (HED) The HED is more complicated. Unlike the VED that excites only the TM waves, an HED will excite both TE and TM waves. For a horizontal electric dipole pointing in the x direction, αˆ =x ˆ; hence, (35.2.1) and (35.2.2) give the TM and the TE components, in anticipation of their plane wave expansions, as iI` ∂2 eikr E = , (35.2.8) z 4πω ∂z∂x r I` ∂ eikr H = − . (35.2.9) z 4π ∂y r 10One can perform a sanity check on the odd and even symmetry of the fields’ z-component by sketching the fields of a static horizontal electric dipole. 11See Chew, Waves and Fields in Inhomogeneous Media [35], p. 46, (2.1.6) and (2.1.7). Or we can gather from (14.1.6) to (14.1.7) that the µiHiz transmits like Eiy at a dielectric interface, and by duality, iEiz transmits like Hiy. 392 Electromagnetic Field Theory Then, with the Sommerfeld identity (35.1.16), we can expand the above as ¢ ∞ iI` E = ± cos φ dk k2J (k ρ)eikz |z| (35.2.10) z 4πω ρ ρ 1 ρ 0 ¢ ∞ I` k2 ρ ikz |z| Hz = i sin φ dkρ J1(kρρ)e . (35.2.11) 4π kz 0 Now, Equation (35.2.10) represents the wave expansion of the TM field, while (35.2.11) repre- sents the wave expansion of the TE field in terms of Sommerfeld integrals which are plane-wave expansions in disguise. Observe that because Ez is odd about z = 0 in (35.2.10), the down- going wave has an opposite sign from the upgoing wave. At this point, the above are just the primary field generated by the source. On top of a stratified medium, the downgoing wave is reflected accordingly, depending on its wave type. Consequently, we have ¢ ∞ iI` h i 2 ik1z |z| ˜TM ik1z (z+2d1) E1z = cos φ dkρ kρJ1(kρρ) ±e − R12 e , (35.2.12) 4πω1 0 ¢ ∞ 2 iI` kρ h i ik1z |z| ˜TE ik1z (z+2d1) H1z = sin φ dkρ J1(kρρ) e + R12 e . (35.2.13) 4π k1z 0 ˜TM Notice that the negative sign in front of R12 in (35.2.12) follows because the downgoing wave in the primary field has a negative sign as shown in (35.2.10). 35.3 Stationary Phase Method—Fermat’s Principle Sommerfeld integrals are rather complex, and by themselves, they do not offer much physical insight into the physics of the field. To elucidate the physics, we can apply the stationary phase method to find approximations of these integrals when the frequency is high, or kr is large, or the observation point is many wavelengths away from the source point. It turns out that this method is initmately related to Fermat’s principle. In order to avoid having to work with special functions like Bessel functions, we convert the Sommerfeld integrals back to spectral integrals in the cartesian coordinates. We could have obtained the aforementioned integrals in cartesian coordinates were we to start with the Weyl identity instead of the Sommerfeld identity. To do the back conversion, we make use of the identity, ¤∞ ¢ ∞ eik0r i eikxx+iky y+ikz |z| k ρ ikz |z| = dkxdky = i dkρ J0(kρρ)e . (35.3.1) r 2π kz kz −∞ 0 Spectral Expansions of Source Fields—Sommerfeld Integrals 393 We can just focus our attention on the reflected wave term in (35.2.6) and rewrite it in cartesian coordinates to get ¤∞ −I` k2 + k2 R x y TM ikxx+iky y+ik1z (z+2d1) E1z = 2 dkxdky R12 e 8π ω1 k1z −∞ ¤∞ 1 ikxx+iky y+ik1z (z+2d1) = dkxdky F (kx, ky)e (35.3.2) k1z −∞ where −I` 2 2 TM F (kx, ky) = 2 (kx + ky)R12 8π ω1 2 2 2 2 In the above, kx +ky +k1z = k1 is the dispersion relation satisfied by the plane wave in region q TM 2 2 2 1. Also, R12 is dependent on kiz = ki − kx − ky in cartesian coordinates, where i = 1, 2. Now the problem reduces to finding the approximation of the following integral: ¤∞ 1 R irh(kx,ky ) E1z = dkxdky F (kx, ky)e (35.3.3) k1z −∞ where x y z rh(k , k ) = r k + k + k , (35.3.4) x y x r y r 1z r We want to approximate the above integral when kr is large. For simplicity, we have set d1 = 0 to begin with. 394 Electromagnetic Field Theory Figure 35.6: In this figure, t can represent kx or ky when one of them is varying. Around the stationary phase point, the function h(t) is slowly varying. In this figure, λ = r, and iλh(kx,ky ) irh(kx,ky ) g(kx, ky, λ) = e = e . When λ = r is large, the function g(λ, kx, ky) is rapidly varying with respect to either kx or ky. Hence, most of the contributions to the integral comes from around the stationary phase point. irh(kx,ky ) In the above, e is a rapidly varying function of kx and ky when x, y, and z are 12 large, or r is large compared to wavelength. In other words, a small change in kx or ky will cause a large change in the phase of the integrand, or the integrand will be a rapidly varying function of kx and ky. Due to the cancellation of the integral when one integrates a rapidly varying function, most of the contributions to the integral will come from around the stationary point of h(kx, ky) or where the function is least slowly varying. Otherwise, the integrand is rapidly varying away from this point, and the integration will destructively cancel with each other, while around the stationary point, they will add constructively. The stationary point in the kx and ky plane is found by setting the derivatives of h(kx, ky) with respect to to kx and ky to zero. By so doing ∂h x k z ∂h y k z = − x = 0, = − y = 0 (35.3.5) ∂kx r k1z r ∂ky r k1z r The above represents two equations from which the two unknowns, kxs and kys, at the stationary phase point can be solved for. By expressing the above in spherical coordinates, x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, the values of (kxs, kys), that satisfy the above equations are kxs = k1 sin θ cos φ, kys = k1 sin θ sin φ (35.3.6) with the corresponding k1zs = k1 cos θ. 12The yardstick in wave physics is always wavelength. Large distance is also synonymous to increasing the frequency or reducing the wavelength. Spectral Expansions of Source Fields—Sommerfeld Integrals 395 When one integrates on the kx and ky plane, the dominant contribution to the integral will come from the point in the vicinity of (kxs, kys). Assuming that F (kx, ky) is slowly varying, we can equate F (kx, ky) to a constant equal to its value at the stationary phase point, and say that ¤∞ 1 eik1r R ikxx+iky y+ik1z z E1z ' F (kxs, kys) e dkxdky = 2πF (kxs, kys) (35.3.7) k1z ir −∞ In the above, the integral can be performed in closed form using the Weyl identity. The above expression has two important physical interpretations. (i) Even though a source is emanating plane waves in all directions in accordance to (35.1.12), at the observation point r far away from the source point, only one or few plane waves in the vicinity of the stationary phase point are important. They interfere with each other constructively to form a spherical wave that represents the ray connect- ing the source point to the observation point. Plane waves in other directions interfere with each other destructively, and are not important. That is the reason that the source point and the observation point is connected only by one ray, or one bundle of plane waves in the vicinity of the stationary phase point. These bundle of plane waves are also almost paraxial with respect to each other. (ii) The function F (kx, ky) could be a very complicated function like the reflection coefficient TM R , but only its value at the stationary phase point matters. If we were to make d1 6= 0 again in the above analysis, the math remains similar except that now, we replace r p 2 2 2 with rI = x + y + (z + 2d1) . Due to the reflecting half-space, the source point has an image point as shown in Figure 35.7 This physical picture is shown in the figure where rI now is the distance of the observation point to the image point. The stationary phase method extract a ray that emanates from the source point, bounces off the half- space, and the reflected ray reaches the observer modulated by the reflection coefficient RTM . But the value of the reflection coefficient that matters is at the angle at which the incident ray impinges on the half-space. (iii) At the stationary point, the ray is formed by the k-vector where k =xk ˆ 1 sin θ cos φ + ykˆ 1 sin θ sin φ +zk ˆ 1 cos theta. This ray points in the same direction as the position vector of the observation point r =xr ˆ sin θ cos φ +yr ˆ sin θ sin φ +zr ˆ cos theta. In other words, the k-vector and the r-vector point in the same direction. This is reminiscent of Fermat principle, because when this happens, the ray propagates with the minimum phase between the source point and the observation point. When z → z + 2d1, the ray for the image source is altered to that shown in Figure 35.7 where it the ray is minimum phase from the image source to the obervation point. Hence, the stationary phase method is initimately related to Fermat’s principle. 396 Electromagnetic Field Theory Figure 35.7: At high frequencies, the source point and the observation point are connected by a ray. The ray represents a bundle of plane waves that interfere constructively. This even true for a bundle of plane waves that reflect off an interface. So ray theory or ray optics prevails here, and the ray bounces off the interface according to the reflection coefficient of a plane wave impinging at the interface with θI . 35.4 Riemann Sheets and Branch Cuts13 The Sommerfeld integrals will have integrands that are multi-value or double value. Proper book keeping is needed so that the evaluation of these integrals can be performed unambigu- 2 2 1/2 ously. The function kz = (k0 − kρ) in (35.1.15) and (35.1.16) are double-value functions because, in taking the square root of a number, two values are possible. In particular, kz is a double-value function of kρ. Consequently, for every point on a complex kρ plane in Figure 35.4, there are two possible values of kz. Therefore, as an example, the integral (35.1.11) is undefined unless we stipulate which of the two values of kz is adopted in performing the integration. A multivalue function is denoted on a complex plane with the help of Riemann sheets [35, 89]. For instance, a double-value function such as kz is assigned two Riemann sheets to a single complex plane. On one of these Riemann sheets, kz assumes a value just opposite in sign to the value on the other Riemann sheet. The correct sign for kz is to pick the square root solution so that =m(kz) > 0. This will ensure a decaying wave from the source. 35.5 Some Remarks14 Even though we have arrived at the solutions of a point source on top of a layered medium by heuristic arguments of plane waves propagating through layered media, they can also be derived more rigorously. For example, Equation (35.2.6) can be arrived at by matching boundary conditions at every interface. The reason why a more heuristic argument is still valid is due to the completeness of Fourier transforms. It is best explained by putting a source over a half space and a scalar problem. 13This may be skipped on first reading. 14This may be skipped on first reading. Spectral Expansions of Source Fields—Sommerfeld Integrals 397 We can expand the scalar field in the upper region as ¤∞ ikxx+iky y Φ1(x, y, z) = dkxdkyΦ˜ 1(kx, ky, z)e (35.5.1) −∞ and the scalar field in the lower region as ¤∞ ikxx+iky y Φ2(x, y, z) = dkxdkyΦ˜ 2(kx, ky, z)e (35.5.2) −∞ If we require that the two fields be equal to each other at z = 0, then we have ¤∞ ¤∞ ikxx+iky y ikxx+iky y dkxdkyΦ˜ 1(kx, ky, z = 0)e = dkxdkyΦ˜ 2(kx, ky, z = 0)e (35.5.3) −∞ −∞ In order to remove the integral, and replace it with a simple scalar problem, one has to impose the above equation for all x and y. Then by the completeness of Fourier transform implies that15 Φ˜ 1(kx, ky, z = 0) = Φ˜ 2(kx, ky, z = 0) (35.5.4) The above equation is much simpler than that in (35.5.3). In other words, due to the com- pleteness of Fourier transform, one can match a boundary condition spectral-component by spectral-component. If the boundary condition is matched for all spectral components, than (35.5.3) is also true. 15Or that we can perform a Fourier inversion on the above integrals. 398 Electromagnetic Field Theory Lecture 36 Computational Electromagnetics, Numerical Methods Due to the advent of digital computers and the blinding speed at which computations can be done, numerical methods to seek solutions of Maxwell’s equations have become vastly popular. Massively parallel digital computers now can compute at breakneck speed of tera\peta\exa- flops throughputs [219], where FLOPS stands for “floating operations per second”. They have also spawn terms that we have not previously heard of (see also Figure 36.1). 399 400 Electromagnetic Field Theory Figure 36.1: Nomenclature for measuring the speed of modern day computers (courtesy of Wikipedia [219]). We repeat a quote from Freeman Dyson—“Technology is a gift of God. After the gift of life it is perhaps the greatest of God’s gifts. It is the mother of civilizations, of arts and of sciences.” The spurr for computer advancement is due to the second world war. During then, men went to war while women stayed back to work as computers, doing laborious numerical computations manually (see Figure 36.2 [220]): The need for a faster computer is obvious. Unfortunately, in the last half century or so, we have been using a large part of the gift of technology to destroy God’s greatest gift, life, in warfare! Figure 36.2: A woman working as a computer shortly after the second world war (courtesy of Wikipedia [220]). Computational Electromagnetics, Numerical Methods 401 36.1 Computational Electromagnetics, Numerical Meth- ods Due to the high fidelity of Maxwell’s equations in describing electromagnetic physics in nature, and they have been validated to high accuracy (see Section 1.1), often time, a numerical solution obtained by solving Maxwell’s equations is more reliable than laboratory experiments. This field is also known as computational electromagnetics. Numerical methods exploit the blinding speed of modern digital computers to perform calculations, and hence to solve large system of equations. Computational electromagnetics consists mainly of two classes of numerical solvers: one that solves differential equations directly: the differential-equation solvers; and one that solves integral equations: the integral equation solvers. Both these classes of equations are derived from Maxwell’s equations. 36.2 Examples of Differential Equations An example of differential equations written in terms of sources are the scalar wave equation: (∇2 + k2(r)) φ(r) = Q(r), (36.2.1) An example of vector differential equation for vector electromagnetic field is ∇ × µ −1 · ∇ × E(r) − ω2(r) · E(r) = iωJ(r) (36.2.2) These equations are linear equations, but for inhomogeneous media where k2(r) and (r) are functions of position vector r, generally, they do not have closed form solutions. They have one commonality, i.e., they can be abstractly written as L f = g (36.2.3) where L is the differential operator which is linear, and f is the unknown, and g is the driving source. Differential equations, or partial differential equations, as mentioned before, have to be solved with boundary conditions. Otherwise, there is no unique solution to these equations. In the case of the scalar wave equation (36.2.1), L = (∇2 + k2) is a differential operator. In the case of the electromagnetic vector wave equation (36.2.2), L = (∇×µ −1 ·∇×)−ω2·. Furthermore, f will be φ(r) for the scalar wave equation (36.2.1), while it will be E(r) in the case of vector wave equation for an electromagnetic system (36.2.2). The g on the right-hand side can represent Q in (36.2.1) or iωJ(r) in (36.2.2). 402 Electromagnetic Field Theory 36.3 Examples of Integral Equations Figure 36.3: Geometry for the derivation of the volume-integral equation for scalar waves. The wavenumber k(r) is assumed to be inhomogeneous and hence, a function of position r inside the scatterer, but a constant k0 outside the scatterer. 36.3.1 Volume Integral Equation This course is replete with PDE’s, but we have not come across too many integral equations as yet. In integral equations, the unknown is embedded in the integral. The simplest integral equation to derive is the volume integral equation. Hence, we shall first derive the volume integral equation for the scalar wave case. In this case, the pertinent scalar wave equation is [∇2 + k2(r)]φ(r) = Q(r), (36.3.1) 2 2 2 where k (r) represents an inhomogeneous medium over a finite domain V , and k = k0, which is constant outside V (see Figure 36.3). Next, we define a Green’s function satisfying 2 2 0 0 [∇ + k0]g(r, r ) = −δ(r − r ). (36.3.2) Then, (36.3.1) can be rewritten as 2 2 2 2 [∇ + k0]φ(r) = Q(r) − [k (r) − k0]φ(r). (36.3.3) Note that the right-hand side of (36.3.3) can be considered an equivalent source. Since the Green’s function corresponding to the differential operator on the left-hand side of (36.3.3) is known, by the principle of linear superposition, we can write the formal solution to (36.3.3) as ¢ ¢ 0 0 0 0 0 2 0 2 0 φ(r) = − dV g(r, r )Q(r ) + dV g(r, r )[k (r ) − k0]φ(r ). (36.3.4) Vs V The first term on the right-hand side is just the field due to the source in the absence of the inhomogeneity or the scatterer, and hence, is the incident field. The second term is a Computational Electromagnetics, Numerical Methods 403 2 0 2 volume integral over the space where k (r ) − k0 6= 0, or inside the inhomogeneous scatterer. Therefore, (36.3.4) becomes ¢ 0 0 2 0 2 0 φ(r) = φinc(r) + dV g(r, r )[k (r ) − k0]φ(r ). (36.3.5) V In the above equation, if the total field φ(r0) inside the volume V is known, then φ(r) can be calculated everywhere. But φ(r) is unknown at this point. To solve for φ(r), an integral equation has to be formulated for φ(r). To this end, we imposed (36.3.5) for r in V . Then, φ(r) on the left-hand side and on the right-hand side are the same unknown defined over the same domain. Consequently, (36.3.5) becomes the desired integral equation after rearrangement as ¢ 0 0 2 0 2 0 φinc(r) = φ(r) − dV g(r, r )[k (r ) − kb ]φ(r ), r ∈ V. (36.3.6) V In the above, the unknown φ(r) is defined over a volume V , over which the integration is performed, and hence the name, volume integral equation. Alternatively, the above can be rewritten as 0 0 φinc(r) = φ(r) − G(r, r )φ(r ), r ∈ V, (36.3.7) where G is the integral operator in (36.3.6).1 It is also a Fredholm integral equation of the second kind because the unknown is both inside and outside the integral operator. In the above, integration over repeated variable r0 is implied. Nevertheless, it can be written as L f = g (36.3.8) where L is a linear operator, while f represents the unknown function φ(r) and g is the known function φinc(r). 1Sometimes, this is called the kernel of the integral equation. 404 Electromagnetic Field Theory 36.3.2 Surface Integral Equation Figure 36.4: Geometry for the derivation of the surface-integral equation for vector electro- magnetics waves. (a) The original electromagnetics scattering problem. (b) The equivalent electromagnetics problem by invoking equivalence principle. The surface integral equation method is rather popular in a number of applications, because it employs a homogeneous-medium Green’s function which is simple in form, and the unknowns reside on a surface rather than in a volume.2 The surface integral equation for vector electromagnetic field can be derived using the equivalence theorem also called the Love’s equivalence theorem [182]. Given a scattering problem shown in Figure 36.4(a), it can be replaced by an equivalence problem as shown in Figure 36.4(b). One can verify this by performing a Gedanken experiment as we have done for the other equivalence problems discussed in Section 31.1. In this figure, the total field outside the scatterer, E = Einc + Esca and H = Hinc + Hsca. The impressed equivalence currents are given by Ms = E × nˆ, and Js =n ˆ × H. These impressed currents, together generate the scattered fields outside the scatterer, while they generate zero field inside the scatterer! One can verify that this is the case by performing Gedanken experiments. As such, the scattered fields outside the scatterer can be found from the radiation of the impressed currents Ms and Js. Notice that these currents are radiating via the free-space Green’s function because the scatterer has been removed in this equivalence problem. Now that if the scatterer is a PEC, then the tangential component of the total electric field is zero on the PEC surface. Therefore, Ms = 0 and only Js is radiating via the free-space Green’s function. Note that this equivalence problem is very different from that of an impressed currents on the PEC scatterer as discussed in Section 31.2. There, only the magnetic surface current 2These are sometimes called boundary integral equations method [221, 222]. Computational Electromagnetics, Numerical Methods 405 is radiating in the presence of the PEC, and the Green’s function is that of a current source radiating in the presence of the PEC scatterer, and it is not the free-space Green’s function. Now we can write the fields outside the scatterer using (31.4.17) 1 1 E (r) = ∇×∇× dS0 g(r − r0)n ˆ0 ×H(r0) = ∇×∇× dS0 g(r − r0) J (r0) (36.3.9) sca iω iω s S S In the above, we have swapped r0 and r compared to (31.4.17). Also, we have kept only the electric current Js(r) due ton ˆ × H(r). If we impose the boundary condition that the tangential component of the total electric field is zero, then we arrive atn ˆ ×Esca = −nˆ ×Einc 1 −nˆ × E (r) =n ˆ × ∇×∇× dS0 g(r − r0) J (r0). r ∈ S (36.3.10) inc iω s S In the above,n ˆ × Einc(r) is known on the left hand side on the scatterer’s surface, while the right-hand side has embedded in it the unknown surface current Js(r) =n ˆ × H(r) on the surface the scatter. Therefore, the above is an integral equation for the unknown surface current Js. It can be written as a form of L f = g just like other linear operator equations. 36.4 Function as a Vector Several linear operator equations have been derived in the previous sections. They are all of the form L f = g (36.4.1) In the above, f is a functional vector which is the analogy of the vector f in matrix theory or linear algebra. In linear algebra, the vector f is of length N in an N dimensional space. It can be indexed by a set of countable index, say i, and we can described such a vector with N numbers such as fi, i = 1,...,N explicitly. This is shown in Figure 36.5(a). A function f(x), however, can be thought of as being indexed by x in the 1D case. However, the index in this case is a continuum, and countably infinite. Hence, it corresponds to a vector of infinite dimension and it lives in an infinite dimensional space.3 To make such functions economical in storage, for instance, we replace the function f(x) by its sampled values at N locations, such that f(xi), i = 1,...,N. Then the values of the 4 function in between the stored points f(xi) can be obtained by interpolation. Therefore, a function vector f(x), even though it is infinite dimensional, can be approximated by a finite length vector, f. This concept is illustrated in Figure 36.5(b) and (c). This concept can be generalized to a function of 3D space f(r). If r is sampled over a 3D volume, it can provide an index to a vector fi = f(ri), and hence, f(r) can be thought of as a vector as well. 3When these functions are square integrable implying finite “energy”, these infinite dimensional spaces are called Hilbert spaces. 4 This is in fact how special functions like sin(x), cos(x), exp(x), Jn(x), Nn(x), etc, are computed and stored in modern computers. 406 Electromagnetic Field Theory Figure 36.5: A function can be thought of as a vector. (a) A continuum function f(x) plotted as a function of x. (b) A digitally sampled values of the same function. (c) When stored in a computer, it will be stored as an array vector. 36.5 Operator as a Map 36.5.1 Domain and Range Spaces An operator like L above can be thought of as a map or a transformation. It maps a function f defined in a Hilbert space V to g defined in another Hilbert space W . Mathematically, this is written as L : V → W (36.5.1) indicating that L is a map of vectors in the space V to vectors in the space W . Here, V is also called the domain space (or domain) of L while W is the range space (or range) of L . Computational Electromagnetics, Numerical Methods 407 36.6 Approximating Operator Equations with Matrix Equa- tions 36.6.1 Subspace Projection Methods One main task of numerical method is first to approximate an operator equation L f = g by a matrix equation L · f = g. To achieve the above, we first let N ∼ X f = anfn = g (36.6.1) n=1 In the above, fn, n, . . . , N are known functions called basis functions. Now, an’s are the new unknowns to be sought. Also the above is an approximation, and the accuracy of the approximation depends very much on the original function f. A set of very popular basis functions are functions that form a piece-wise linear interpolation of the function from its nodes. These basis functions are shown in Figure 36.6. Figure 36.6: Examples of basis function in (a) one dimension, (b) two dimension. Each of these functions are define over a finite domain. Hence, they are also called sub-domain basis functions. They can be thought of as interpolatory functions. Upon substituting (36.6.1) into (36.4.1), we obtain N X anL fn = g (36.6.2) n=1 Then, upon multiplying (36.6.2) by wm and integrating over the space that wm(r) is defined, 408 Electromagnetic Field Theory then we have N X an hwm, L fni = hwm, gi , m = 1,...,N (36.6.3) n=1 In the above, the inner product is defined as ¢ hf1, f2i = drf1(r)f2(r) (36.6.4) where the integration is over the support of the functions, or the space over which the functions are defined.5 For PDEs these functions are defined over a 3D coordinate space, while in SIEs, these functions are defined over a surface. In a 1D problems, these functions are defined over a 1D coordinate space. 36.6.2 Dual Spaces The functions wm, m = 1,...,N is known as the weighting functions or testing functions. The testing functions should be chosen so that they can approximate well a function that lives in the range space W of the operator L . Such set of testing functions lives in the dual space of the range space. For example, if fr lives in the range space of the operator L , the set of function fd, such that the inner product hfd, fri exists, forms the dual space of W . 36.6.3 Matrix and Vector Representations The above equation (36.6.3) is a matrix equation of the form L · a = g (36.6.5) where L = hwm, L fni mn (36.6.6) [a]n = an, [g]m = hwm, gi What has effectively happened here is that given an operator L that maps a function that lives in an infinite dimensional Hilbert space V , to another function that lives in another infinite dimensional Hilbert space W , via the operator equation L f = g, we have approximated the Hilbert spaces with finite dimensional spaces (subspaces), and finally, obtain a finite dimensional matrix equation that is the representation of the original infinite dimensional operator equation. This is the spirit of the subspace projection method. In the above, L is the matrix representation of the operator L in the subspaces, and a and g are the vector representations of f and g, respectively, in their respective subspaces. When such a method is applied to integral equations, it is usually called the method of moments (MOM). (Surface integral equations are also called boundary integral equations (BIEs) in other fields [222].) When finite discrete basis are used to represent the surface unknowns, it is also called the boundary element method (BEM) [223]. But when this method is applied to solve PDEs, it is called the finite element method (FEM) [224–227], which is a rather popular method due to its simplicity. 5This is known as the reaction inner product [35,50,131].¡ As oppose to most math and physics literature, ∗ the energy inner product is used [131] where hf1, f2i = drf1 (r)f2(r). Computational Electromagnetics, Numerical Methods 409 36.6.4 Mesh Generation In order to approximate a function defined on an arbitrary shaped surface or volume by a finite sum of basis functions, it is best to mesh (tessellate or discretize) the surface and volume by meshes. In 2D, all shapes can be tessellated by unions of triangles, while a 3D volume can be meshed (tessellated) by unions of tetrahedrons. Such meshes are used not only in CEM, but in other fields such as solid mechanics. Hence, there are many “solid modeling” commercial software available to generate sophisticated meshes. When a surface is curved, or of arbitrary shape, it can be meshed by union of triangles as shown in Figure 36.7. When a volume is of arbitrary shape or a volume is around an arbitrary shape object, it can be meshed by tetrahedrons as shown in Figure 36.8. Then basis functions as used in (36.6.1) are defined to interpolate the field between nodal values or values defined on the edges of a triangle or a tetrahedron. Figure 36.7: An arbitrary surface can be meshed by a union of triangles. Figure 36.8: A volume region can be meshed by a union of tetrahedra. But the surface of the aircraft is meshed with a union of trianlges (courtesy of gmsh.info). 410 Electromagnetic Field Theory 36.6.5 Differential Equation Solvers versus Integral Equation Solvers As have been shown, the two classes of numerical solvers for Maxwell’s equations consist of differential equation solvers and integral equation solvers. Differential equation solvers are generally easier to implement. As shall be shown in the next lecture, they can also be easily implemented using finite difference solver. The unknowns in a differential equation solver are the fields. The fields permeate all of space, and hence, the unknowns are volumetrically distributed. When the fields are digitized by representing them by their point values in space, they require a large number of unknowns to represent. The plus side is that the matrix system associated with a differential equation solver is usually sparse, requiring less storage and less time to solve. As has been shown, integral equation solvers are formulated using Green’s functions. That is integral equations are derived from Maxwell’s equations using Green’s function, where the unknowns now are surface sources such as surface electric and magnetic currents. Therefore, the unknowns are generally smaller, living only on the surface of a scatterer (or they occupy a smaller part of space). Hence, they can be approximated by a smaller set of unknowns. Thus, the matrix systems generally are smaller. Once the currents are found, then the fields they generate can also be computed. Since the derivation of integral equations requires the use of Green’s functions, they are in general singular when r = r0, or when the observation point (observation point) r and the source point r0 coincide. Care has to be taken to discretize the integral equations. Moreover, a Green’s function connects every current source point on the surface of a scatterer with every other source points yielding a dense matrix system. But fast methods have been developed to solve such dense matrix systems [9]. 36.7 Solving Matrix Equation by Optimization Given a matrix equation, there are many ways to seek its solution. The simplest way is to find the inverse of the matrix operator by direct inversions (e.g., using Gaussian elimination [228] or Kramer’s rule [229]) have computational complexity6 of O(N 3), and requiring storage of O(N 2). Due to the poor computational and memory complexity of direct inversion, when N is large, other methods have to be sought. To this end, it is better to convert the solving of a matrix equation into an optimization problem. These methods can be designed so that a much larger system can be solved with an existing resource of a digital computer. Optimization problem results in finding the stationary point of a functional.7 First, we will figure out how to find such a functional. Consider a matrix equation given by L · f = g (36.7.1) 6The scaling of computer time with respect to the number of unknowns (degrees of freedom) is known in the computer parlance as computational complexity. 7Functional is usually defined as a function of a function [35, 49]. Here, we include a function of a vector to be a functional as well. Computational Electromagnetics, Numerical Methods 411 For simplicity, we consider L as a symmetric matrix.8 Then the corresponding functional is I = f t · L · f − 2f t · g (36.7.2) Such a functional is called a quadratic functional because it is analogous to I = Lx2 − 2xg, which is quadratic, in its simplest 1D rendition. Taking the first variation with respect to f, namely, we let f = fo +δf. Then we substitute this into the above, and collect the leading order and first order terms. Then we find the first order approximation of the functional I as t t t δI = δf · L · fo + fo · L · δf − 2δf · g (36.7.3) If L is symmetric, the first two terms are the same, and the above becomes t t δI = 2δf · L · fo − 2δf · g (36.7.4) For fo to be the optimal point or the stationary point, then its first variation has to be zero, or that δI = 0. Thus we conclude that at the optimal point (or the stationary point), L · fo = g (36.7.5) Hence, the optimal point to the functional I in (36.7.2) is the solution to (36.7.1) or (36.7.5). 36.7.1 Gradient of a Functional The above method, when applied to an infinite dimensional Hilbert space problem, is called variational method, but the main ideas are similar. The wonderful idea about such a method is that instead of doing direct inversion of a matrix system (which is expensive), one can search for the optimal point or stationary point of the quadratic functional using gradient search or gradient descent methods or some optimization method. It turns out that the gradient of a quadratic functional can be found quite easily. Also it is cheaper to compute the gradient of a functional than to find the inverse of a matrix operator. To do this, it is better to write out functional using index (or indicial, or Einstein) notation [230]. In this notation, the functional first variation δI in (36.7.4) becomes δI = 2δfjLijfi − 2δfjgj (36.7.6) Also, in this notation, the summation symbol is dropped, and summations over repeated indices are implied. In the above, we neglect to distinguish between fo and f. It is implied that f represents the optimal point. In this notation, it is easier to see what a functional derivative is. We can differentiate the above with respect to fj easily to arrive at ∂I = 2Lijfi − 2gj (36.7.7) ∂fj 8Functional for the asymmetric case can be found in Chew, Waves and Fields in Inhomogeneous Media, Chapter 5 [35]. 412 Electromagnetic Field Theory Notice that the remaining equation has one index j remaining in index notation, meaning that it is a vector equation. We can reconstitute the above using our more familiar matrix notation that δI = ∇ I = 2L · f − 2g (36.7.8) δf f The left-hand side is a notation for the functional derivative or the gradient of a functional in a multi-dimensional space whichi is a vector obviated by indicial notation. And the right-hand side is the expression for calculating this gradient. One needs only to perform a matrix-vector product to find this gradient. Hence, the computational complexity of finding this gradient is O(N 2) at worst if L is a dense matrix, and as low as O(N) if L is a sparse matrix.9 In a gradient search method, such a gradient is calculated repeatedly until the optimal point is found. Such methods are called iterative methods. α If the optimal point can be found in Niter iterations, then the CPU time scales as NiterN where 1 < α < 2. There is a clever gradient search algorithm, called the conjugate gradient method that can find the exact optimal point in Niter = N in exact arithmetics. But exact solution is not needed in an optimal solution: an approximate solution suffices. In many gradient search solutions, to obtain an approximate solution where the error is acceptable, α α 3 Niter N. The total solution time or solve time which is NiterN NN N , resulting in great savings in computer time, especially if α = 1. This is the case for FEM [227], [231], [232], [233], [226], and fast multipole algorithm [234], [235]. What is more important is that this method does not require the storage of the matrix L, but a computer code that produces the vector go = L · f as an output, with f as an input. Both f and go require only O(N) memory storage. Such methods are called matrix-free methods. Even when L is a dense matrix, which is the case if it is the matrix representation of matrix representation of some Green’s function, fast methods now exist to perform the dense matrix-vector product in O(N log N) operations.10 The value I is also called the cost function, and its minimum is sought in the seeking of the solution by gradient search methods. Detail discussion of these methods is given in [236]. Figure 36.9 shows the contour plot of a cost function in 2D. When the condition number11 of the matrix L is large (implying that the matrix is ill-conditioned), the contour plot will resemble a deep valley. And hence, the gradient search method will tend to zig-zag along the way as it finds the optimal solution. Therefore, convergence is slow for matrices with large condition numbers 9This is the case for many differential equation solvers such as finite-element method or finite-difference method. 10Chew et al, Fast and Efficient Algorithms in CEM [9]. 11This is the ratio of the largest eigenvalue of the matrix to its smallest eigenvalue. Computational Electromagnetics, Numerical Methods 413 Figure 36.9: Plot of a 2D cost function, I(x, y) for an ill-conditioned system (courtesy of Numerical Recipe [236]). A higher dimensional plot of this cost function will be difficult. Figure 36.10 shows a cartoon picture in 2D of the histories of different search paths from a machine-learning example where a cost functional similar to I has to be minimized. Finding the optimal point or the minimum point of a general functional is still a hot topic of research: it is important in artificial intelligence. Figure 36.10: Gradient search or gradient descent method is finding an optimal point (courtesy of Y. Ioannou: https://blog.yani.io/sgd/). 414 Electromagnetic Field Theory Lecture 37 Finite Difference Method, Yee Algorithm In this lecture, we are going to introduce one of the simplest methods to solve Maxwell’s equations numerically. This is the finite-difference time-domain method first proposed by Yee [237] and popularized by Taflove [238]. Because of its simplicity, and that a simple Maxwell solver can be coded in one afternoon, almost every physics or electrical engineering laboratory has a home-grown version of the finite-difference time-domain solver. This method is the epitome of that “simplicity rules.”1 Professor Hermann Haus at MIT used to say: find the simplest method to do things. Complicated methods will be forgotten, but the simplest method will prevail. This is also reminiscent of Einstein’s saying, “Everything should be made as simple as possible, but no simpler!” 37.1 Finite-Difference Time-Domain Method To obtain the transient (time-domain) solution of the wave equation for a more general, inhomogeneous medium, a numerical method has to be used. The finite-difference time- domain (FDTD) method, a numerical method, is particularly suitable for solving transient problems. Moreover, it is quite versatile, and given the present computer technology, it has been used with great success in solving many practical problems. This method is based on a simple Yee algorithm [237] and has been vastly popularized by Taflove [238,239]. In the finite-difference method, continuous space-time is replaced with a discrete space- time. Then, in the discrete space-time, partial differential equations are replaced with finite difference equations. These finite difference equations are readily implemented on a digital computer. Furthermore, an iterative or time-stepping scheme can be implemented without having to solve large matrices, resulting in great savings in computer time. Moreover, the matrix for the system of equations is never generated making this a matrix-free method. There is no need for matrix management as one writes this numerical solver. More recently, 1“rule” is used as a verb. 415 416 Electromagnetic Field Theory the development of parallel processor architectures in computers has also further enhanced the efficiency of the finite-difference time-domain scheme [240]. The finite-difference method is also described in numerous works (see, for example, Potter 1973 [241]; Taflove 1988 [238]; Ames 2014 [242]; and Morton 2019 [243]. 37.1.1 The Finite-Difference Approximation Consider first a scalar wave equation of the form 1 ∂2 φ(r, t) = µ(r)∇ · µ−1(r)∇φ(r, t). (37.1.1) c2(r) ∂t2 The above equation appears in scalar acoustic waves or a 2D electromagnetic waves in inho- mogeneous media [35,244]. To convert the above into a form that can be solved by a digital computer easily, first, one needs to find finite-difference approximations to the time derivatives. Then, the time derivative can be approximated in many ways. For example, a derivative can be approximated by forward, backward, and central finite difference formulas [245]. ∂φ(r, t) φ(r, t + ∆t) − φ(r, t) Forward difference: ≈ , (37.1.2) ∂t ∆t ∂φ(r, t) φ(r, t) − φ(r, t − ∆t) Backward difference: ≈ , (37.1.3) ∂t ∆t ∂φ(r, t) φ(r, t + ∆t ) − φ(r, t − ∆t ) Central difference: ≈ 2 2 , (37.1.4) ∂t ∆t Finite Difference Method, Yee Algorithm 417 Figure 37.1: Different finite-difference approximations for the time derivative. One can eye- ball that the central difference formula is the best. This can be further confirmed by a Taylor series analysis. where ∆t is a small number. Of the three methods of approximating the time derivative, the central-difference scheme is the best approximation, as is evident from Figure 37.1. The errors in the forward and backward differences are O(∆t) (or first-order error) while the central- difference approximation has an error O[(∆t)2] (or second-order error). This can be easily verified by Taylor series expanding the right-hand sides of (37.1.2) to (37.1.4). Consequently, using the central-difference formula twice, we arrive at " # ∂2 ∂ φ(r, t + ∆t ) − φ(r, t − ∆t ) φ(r, t) ≈ 2 2 (37.1.5) ∂t2 ∂t ∆t φ(r, t + ∆t) − 2φ(r, t) + φ(r, t − ∆t) ≈ . (37.1.6) (∆t)2 418 Electromagnetic Field Theory Next, if the function φ(r, t) is indexed on discrete time steps on the t axis, such that for t = l∆t, then φ(r, t) = φ(r, l∆t) = φl(r), where l is an integer is used to count the time steps. Using this notation, Equation (37.1.6) then becomes ∂2 φl+1(r) − 2φl(r) + φl−1(r) φ(r, t) ≈ . (37.1.7) ∂t2 (∆t)2 37.1.2 Time Stepping or Time Marching With this notation and approximations, (37.1.1) can be approximated by a time-stepping (or time-marching) formula, namely, φl+1(r) =∼ c2(r)(∆t)2µ(r)∇ · µ−1(r)∇φl(r) + 2φl(r) − φl−1(r). (37.1.8) Therefore, given the knowledge of φ(r, t) at t = l∆t and t = (l−1)∆t for all r, one can deduce φ(r, t) at t = (l +1)∆t. In other words, given the initial values of φ(r, t) at, for example, t = 0 and t = ∆t, φ(r, t) can be deduced for all subsequent times, provided that the time-stepping formula is stable. At this point, the right-hand side of (37.1.8) involves the space derivatives. There exist a plethora of ways to approximate and calculate the right-hand side of (37.1.8) numerically. Here, we shall illustrate again the use of the finite-difference method to calculate the right- hand side of (37.1.8). Before proceeding further, note that the space derivatives on the right-hand side in cartesian coordinates are ∂ ∂ ∂ ∂ ∂ ∂ µ(r)∇ · µ−1(r)∇φ(r) = µ µ−1 φ + µ µ−1 φ + µ µ−1 φ. (37.1.9) ∂x ∂x ∂y ∂y ∂z ∂z Then, one can approximate, using central differencing that ∂ 1 ∆z ∆z φ(x, y, z) ≈ φ x, y, z + − φ x, y, z − , (37.1.10) ∂z ∆z 2 2 Consequently, using central differencing two times, ∂ ∂ 1 ∆z µ−1 φ(x, y, z) ≈ µ−1 z + φ(x, y, z + ∆z) ∂z ∂z (∆z)2 2 ∆z ∆z − µ−1 z + + µ−1 z − φ(x, y, z) 2 2 ∆z +µ−1 z − φ(x, y, z − ∆z) . (37.1.11) 2 Furthermore, after denoting φ(x, y, z) = φm,n,p, µ(x, y, z) = µm,n,p, on a discretized grid point at x = m∆x, y = n∆y, z = p∆z, we have (x, y, z) = (m∆x, n∆y, p∆z), and then ∂ −1 ∂ 1 h −1 µ φ(x, y, z) ≈ 2 µm,n,p+ 1 φm,n,p+1 ∂z ∂z (∆z) 2 −1 −1 −1 i − µ 1 + µ 1 φm,n,p + µ 1 φm,n,p−1 . (37.1.12) m,n,p+ 2 m,n,p− 2 m,n,p− 2 Finite Difference Method, Yee Algorithm 419 This cumbersome and laborious looking equation can be abbreviated if we define a central difference operator as2 ¯ 1 ∂zφm = φm+ 1 − φm− 1 (37.1.13) ∆z 2 2 Then the right-hand side of the (37.1.12) can be written succinctly as ∂ ∂ µ−1 φ(x, y, z) ≈ ∂¯ µ ∂¯ φ (37.1.14) ∂z ∂z z m,n,p z m,n,p With similar approximations to the other terms in (37.1.9), (37.1.8) is now compactly written as l+1 2 2 ¯ ¯ ¯ ¯ ¯ ¯ φm,n,p = (∆t) cm,n,pµm,n,p ∂xµm,n,p∂x + ∂yµm,n,p∂y + ∂zµm,n,p∂z φm,n,p l l−1 + 2φm,n,p − φm,n,p. (37.1.15) The above can be readily implemented on a computer for time stepping. Notice however, that the use of central differencing results in the evaluation of medium property µ at half grid points. This is inconvenient, as the introduction of material values at half grid points increases computer memory. Hence, it is customary to store the medium property at the integer grid points for ease of book-keeping, and to approximate 1 µm+ 1 ,n,p ' (µm+1,n,p + µm,n,p), (37.1.16) 2 2 µ 1 + µ 1 ' 2µm,n,p, (37.1.17) m+ 2 ,n,p m− 2 ,n,p and so on. Moreover, if µ is a smooth function of space, it is easy to show that the errors in the above approximations are of second order by Taylor series expansions. For a homogeneous medium, with ∆x = ∆y = ∆z = ∆s, namely, we assume the space steps to be equal in all directions. (37.1.15) written explicitly becomes ∆t 2 φl+1 = c2 φl + φl + φl + φl + φl m,n,p ∆s m+1,n,p m−1,n,p m,n+1,p m,n−1,p m,n,p+1 l l l l−1 + φm,n,p−1 − 6φm,n,p + 2φm,n,p − φm,n,p. (37.1.18) l+1 Notice then that with the central-difference approximation, the value of φm,n,p is dependent l l l l l−1 only on φm,n,p, and its nearest neighbors, φm±1,n,p, φm,n±1,p, φm,n,p±1, and φm,n,p, its value at the previous time step. Moreover, in the finite-difference scheme outlined above, no matrix inversion is required at each time step. Such a scheme is also known as an explicit scheme. The use of an explicit scheme is a major advantage of the finite-difference method compared to the finite-element methods. Consequently, in order to update N grid points using (37.1.15) or (37.1.18), O(N) multiplications are required for each time step. In comparison, O(N 3) multiplications are required to invert an N × N full matrix, e.g., using Gaussian elimination. The simplicity and efficiency of these finite-difference algorithms have made them vastly popular. 2This is in the spirit of [246]. 420 Electromagnetic Field Theory 37.1.3 Stability Analysis The implementation of the finite-difference time-domain scheme using time-marching does not always lead to a stable scheme. Hence, in order for the solution to converge, the time- stepping scheme must at least be stable. Consequently, it is useful to find the condition under which a numerical finite-difference scheme is stable. To do this, one performs the von Neumann stability analysis (von Neumann 1943 [247]) on Equation (37.1.18). We will assume the medium to be homogeneous to simplify the analysis. As shown previously in Section 35.1, a point source gives rise to a spherical wave that can be expanded in terms of sum of plane waves in different directions. It also implies that any wave emerging from sources can be expanded in terms of sum of plane waves. This is the spirit of the spectral expansion method. So if a scheme is not stable for a plane wave, it would not be stable for any wave. Consequently, to perform the stability analysis, we assume a propagating plane wave as a trial solution φ(x, y, z, t) = A(t)eikxx+iky y+ikz z, (37.1.19) In discretized form, by letting ∆x = ∆y = ∆z = ∆s, it is just l l ikxm∆s+iky n∆s+ikz p∆s φm,n,p = A e . (37.1.20) l −iωl∆t We can imagine that A = A0e , so that the above is actually a Fourier plane wave mode in the frequency domain. Using (37.1.20), it is easy to show that for the x space derivative, l l l l φm+1,n,p − 2φm,n,p + φm−1,n,p = 2[cos(kx∆s) − 1]φm,n,p k ∆s = −4 sin2 x φl . (37.1.21) 2 m,n,p The space derivatives in y and z directions can be similarly derived. The time derivative can be similarly approximated, and it is approximately equal to ∂2 φ(r, t)(∆t)2 ≈ φl+1 − 2φl + φl−1 . (37.1.22) ∂t2 m,n,p m,n,p m,n,p Substituting (37.1.20) into the above, we have the second time derivative being proportional to ∂2 φ(r, t)(∆t)2 ≈ (Al+1 − 2Al + Al−1)eikxm∆s+iky n∆s+ikz p∆s (37.1.23) ∂t2 To simplify further, one can assume that Al+1 = gAl. (37.1.24) This is commensurate with assuming that −iωt A(t) = A0e (37.1.25) Finite Difference Method, Yee Algorithm 421 where ω can be complex. In other words, our trial solution (37.1.19) is also a time-harmonic signal where ω can be complex. If the finite-difference scheme is unstable for such a signal, it is unstable for all signals. Consequently, the time derivative is proportional to ∂2 φ(r, t)(∆t)2 ≈ (g − 2 + g−1)φl (37.1.26) ∂t2 m,n,p We need to find the value of g for which the solution (37.1.20) satisfies (37.1.18). To this end, one uses (37.1.21) and (37.1.24) in (37.1.18), and repeating (37.1.21), which is for m variable in the x direction, for the n and p variables in the x and y directions as well, one obtains ∆t 2 k ∆s k ∆s (g − 2 + g−1)φl = −4 c2 sin2 x + sin2 y m,n,p ∆s 2 2 k ∆s + sin2 z φl 2 m,n,p 2 2 l = −4r s φm,n,p, (37.1.27) where ∆t k ∆s k ∆s k ∆s r = c, s2 = sin2 x + sin2 y + sin2 z . (37.1.28) ∆s 2 2 2 l 3 Equation (37.1.27) implies that, for nonzero φm,n,p, g2 − 2g + 4r2s2g + 1 = 0, (37.1.29) or that g = (1 − 2r2s2) ± 2rsp(r2s2 − 1) . (37.1.30) In order for the solution to be stable, it is necessary that |g| ≤ 1. But if r2s2 < 1, (37.1.31) the second term in (37.1.30) is pure imaginary, and |g|2 = (1 − 2r2s2)2 + 4r2s2(1 − r2s2) = 1, (37.1.32) 2 when (37.1.31) is true. Therefore, stability is ensured. Since from (37.1.28), s ≤ 3 for all kx, ky, and kz, from (37.1.31). From (37.1.31), we conclude that 1 r < s But we also know that 1 1 ≥ √ s 3 3For those who are more mathematically inclined, we are solving an eigenvalue problem in disguise. Re- member that a function is a vector, even after it has been discretized:) 422 Electromagnetic Field Theory The above two inequalities will be satisfied if the general condition is 1 ∆s r < √ , or ∆t < √ . (37.1.33) 3 c 3 The above is the general condition for stability. The above analysis is for 3 dimensional problems. It is clear from the above analysis that for an n-dimensional problem where n = 1, 2, 3, then ∆s ∆t < √ . (37.1.34) c n One may ponder on the meaning of this inequality further: but it is only natural that the time step ∆t has to be bounded from above. Otherwise, one arrives at the ludicrous notion that the time step can be arbitrarily large thus violating causality. Moreover, if the grid points of√ the finite-difference scheme are regarded as a simple cubic lattice, then the distance ∆s/ n is also the distance between the closest lattice planes through the simple cubic lattice.√ Notice that the time for the wave to travel between these two lattice planes is ∆s/(c n ). Consequently, the stability criterion (37.1.34) implies that the time step ∆t has to be less than the shortest travel time for the wave between the lattice planes in order to satisfy causality. In other words, if the wave is time-stepped ahead of the time on the right-hand side of (37.1.34), instability ensues. The above is also known as the CFL (Courant, Friedrichs, and Lewy 1928 [248]) stability criterion. It could be easily modified for ∆x 6= ∆y 6= ∆z [239]. The above analysis implies that we can pick a larger time step if the space steps are larger. A larger time step will allow one to complete generating a time-domain response rapidly. However, one cannot arbitrary make the space step large due to grid-dispersion error, as shall be discussed next. 37.1.4 Grid-Dispersion Error When a finite-difference scheme is stable, it still may not be accurate to produce good results due to the errors in the finite-difference approximations. Hence, it is useful to ascertain the errors in terms of the size of the grid and the time step. An easy error to analyze is the grid- dispersion error. In a homogeneous, dispersionless medium, all plane waves propagate with the same phase velocity. However, in the finite-difference approximation, all plane waves will not propagate at the same phase velocity due to the grid-dispersion error. As a consequence, a pulse in the time domain, which is a linear superposition of plane waves with different frequencies, will be distorted if the dispersion introduced by the finite-difference scheme is intolerable. Therefore, to make things simpler, we will analyze the grid-dispersion error in a homogeneous free space medium. To ascertain the grid-dispersion error, we assume that the solution is time-harmonic, or l −iωl∆t that A = A0e in (37.1.20). In this case, the left-hand side of (37.1.27) becomes ω∆t e−iω∆t − 2 + e+iω∆t φl = −4 sin2 φl . (37.1.35) m,n,p 2 m,n,p Finite Difference Method, Yee Algorithm 423 Then, from (37.1.27), it follows that ω∆t sin = rs, (37.1.36) 2 where r and s are given in (37.1.28). Now, (37.1.36) governs the relationship between ω and kx, ky, and kz in the finite-difference scheme, and hence, is a dispersion relation for the approximate solution. The above gives a rather complicated relationship between the frequency ω and the wave numbers kx, ky, and kz. This is the result of the finite-difference approximation of the scalar wave equation. As a sanity check, when the space and time discretizations become very small, we should recover the dispersion relation of homogeneous medium or free space. But if a medium is homogeneous, it is well known that (37.1.1) has a plane-wave solution of the type given by (37.1.19) where q 2 2 2 ω = c kx + ky + kz = c|k| = ck. (37.1.37) where k =xk ˆ x +yk ˆ y +zk ˆ z is the direction of propagation of the plane wave. Defining the phase velocity to be ω/k = c, this phase velocity is isotropic, or the same in all directions. Moreover, it is independent of frequency. But in (37.1.36), because of the definition of s as given by (37.1.28), the dispersion relation between ω and k is not isotropic. This implies that plane waves propagating in different directions will have different phase velocities. Equation (37.1.36) is the dispersion relation for the approximate solution. It departs from Equation (37.1.37), the exact dispersion relation, as a consequence of the finite-difference ap- proximation. This departure gives rise to errors, which are the consequence of grid dispersion. For example, when c is a constant, (37.1.37) states that the phase velocities of plane waves of different wavelengths and directions are the same. However, this is not true for (37.1.36), as shall be shown. Assuming s small, (37.1.36), after using Taylor series expansion, can be written as ω∆t r3s3 = sin−1 rs =∼ rs + . (37.1.38) 2 6 When ∆s is small, using the small argument approximation for the sine function, one obtains from (37.1.28) ∆s s ' (k2 + k2 + k2)1/2 (37.1.39) 2 x y z Equation (37.1.38), by taking the higher-order Taylor expansion of (37.1.38), then becomes ω∆t ∆s ' r (k2 + k2 + k2)1/2 [1 − δ] (37.1.40) 2 2 x y z where (see [35]) 2 4 4 4 2 2 ∆s kx + ky + kz r ∆s 2 2 2 δ = 2 2 2 − (kx + ky + kz ) (37.1.41) 24 kx + ky + kz 24 424 Electromagnetic Field Theory From the above, one can see that if δ = 0, we retrieve the dispersion relation of the homoge- neous free-space medium. So δ is a measure of the departure of the dispersion relation from that of free space due to our finite-difference approximation. A soothing fact is that when ∆s 1, δ is small. Since k is inversely proportional to wavelength λ, then δ in the correction to the above equation is proportional to ∆s2/λ2. Therefore, to reduce the grid dispersion error, it is necessary for δ to be small or to have ∆s2 1. (37.1.42) λ When this is true, using the fact that r = c∆t/∆s, then (37.1.40) becomes ω q ≈ k2 + k2 + k2 . (37.1.43) c x y z which is close to the dispersion relation of free space as indicated in (37.1.37). Consequently, in order for the finite-difference scheme to propagate a certain frequency content accurately, the grid size must be much less than the wavelength of the corresponding frequency. Furthermore, ∆t must be chosen so that the CFL stability criterion is met. Therefore, the rule of thumb is to choose about 10 to 20 grid points per wavelength. Also, for a plane wave propagating as eik·r, an error δk in the vector k gives rise to cumulative error eiδk·r. The larger the distance traveled, the larger the cumulative phase error, and hence the grid size must be smaller in order to arrest such phase error due to the grid dispersion. 37.2 The Yee Algorithm The Yee algorithm (Yee 1966 [237]) is a simple algorithm specially designed to solve vec- tor electromagnetic field problems on a rectilinear grid. The finite-difference time-domain (FDTD) method (Taflov 1988) when applied to solving electromagnetics problems, usually uses this method. To derive it, Maxwell’s equations are first written in cartesian coordinates: ∂B ∂E ∂E − x = z − y , (37.2.1) ∂t ∂y ∂z ∂B ∂E ∂E − y = x − z , (37.2.2) ∂t ∂z ∂x ∂B ∂E ∂E − z = y − x , (37.2.3) ∂t ∂x ∂y ∂D ∂H ∂H x = z − y − J , (37.2.4) ∂t ∂y ∂z x ∂D ∂H ∂H y = x − z − J , (37.2.5) ∂t ∂z ∂x y ∂D ∂H ∂H z = y − x − J . (37.2.6) ∂t ∂x ∂y z Finite Difference Method, Yee Algorithm 425 Before proceeding any further, it is prudent to rewrite the differential equation form of Maxwell’s equations in their integral form. The first equation above can be rewritten as ¤ ∂ − BxdS = E · dl (37.2.7) ∂t ∆S ∆C where ∆S = ∆x∆z. The approximation of this integral form will be applied to the face that is closest to the observer in Figure 37.2. Hence, one can see that the curl of E is proportional to the time-derivative of the magnetic flux through the suface enclosed by ∆C, which is ∆S. One can see this relationship for the other surfaces of the cube in the figure as well: the electric field is curling around the magnetic flux. For the second half of the above equations, one can see that the magnetic fields are curling around the electric flux, but on a staggered grid. These two staggered grids are intertwined with respect to each other. This is the spirit with which the Yee algorithm is written. He was apparently motivated by fluid dynamics when he did the work. Figure 37.2: The assignment of fields on a grid in the Yee algorithm. l After denoting f(m∆x, n∆y, p∆z, l∆t) = fm,n,p, and replacing derivatives with central finite-differences in accordance with Figure 37.2, (37.2.1) becomes h 1 1 i h i 1 l+ 2 l− 2 1 l l Bx,m,n+ 1 ,p+ 1 − Bx,m,n+ 1 ,p+ 1 = Ey,m,n+ 1 ,p+1 − Ey,m,n+ 1 ,p ∆t 2 2 2 2 ∆z 2 2 1 h l l i − Ez,m,n+1,p+ 1 − Ez,m,n,p+ 1 . (37.2.8) ∆y 2 2 where the above formula is evaluated at t = l∆t. Moreover, the above can be repeated for (37.2.2) and (37.2.3). Notice that in Figure 37.2, the electric field is always assigned to the edge center of a cube, whereas the magnetic field is always assigned to the face center of a cube.4 4This algorithm is intimately related to differential forms which has given rise to the area of discrete exterior calculus [249]. 426 Electromagnetic Field Theory In fact, after multiplying (37.2.8) by ∆z∆y, (37.2.8) is also the approximation of the integral forms of Maxwell’s equations when applied at a face of a cube. By doing so, the left-hand side of (37.2.8), by (37.2.7), becomes h 1 1 i l+ 2 l− 2 (∆y∆z/∆t) B 1 1 − B 1 1 , (37.2.9) x,m,n+ 2 ,p+ 2 x,m,n+ 2 ,p+ 2 which is the time variation of the total flux through an elemental area ∆y∆z. Moreover, by summing this flux on the six faces of the cube shown in Figure 37.2, and using the right- hand side of (37.2.8) and its equivalent, it can be shown that the magnetic flux adds up to ∂ zero. Hence, ∂t ∇ · B = 0 condition is satisfied within the numerical approximations of Yee algorithm. The above shows that if the initial value implies that ∇ · B = 0, the algorithm will preserve this condition. So even though we are solving Faraday’s law, Gauss’ law is also satisfied. This is important in maintaining the stability of the numerical algorithm [250]. Furthermore, a similar approximation of (37.2.4) leads to h i h 1 1 i 1 l l−1 1 l− 2 l− 2 Dx,m+ 1 ,n,p − Dx,m+ 1 ,n,p = Hz,m+ 1 ,n+ 1 ,p − Hz,m+ 1 ,n− 1 ,p ∆t 2 2 ∆y 2 2 2 2 h 1 1 i 1 1 l− 2 l− 2 l− 2 − Hy,m+ 1 ,n,p+ 1 − Hy,m+ 1 ,n,p− 1 − Jx,m+ 1 ,n,p. (37.2.10) ∆z 2 2 2 2 2 Also, similar approximations apply for (37.2.5) and (37.2.6). In addition, the above has an interpretation similar to (37.2.8) if one thinks in terms of a cube that is shifted by half a grid point in each direction. Hence, the approximations of (37.2.4) to (37.2.6) are consistent with ∂ the approximation of ∂t ∇ · D = −∇ · J. This way of alternatively solving for the B and D fields in tandem while the fields are placed on a staggered grid is also called the leap-frog scheme. In the above, D = E and B = µH. Since the magnetic field and the electric field are assigned on staggered grids, µ and may have to be assigned on staggered grids. This does not usually lead to serious problems if the grid size is small. Alternatively, (37.1.16) and (37.1.17) can be used to remove this problem, and to reduce storage. By eliminating the E or the H field from the Yee algorithm, it can be shown that the Yee algorithm is equivalent to finite differencing the vector wave equation directly. Hence, the Yee algorithm is also constrained by the CFL stability criterion. The following figures show some results of FDTD simulations. Because the answers are in the time-domain, beautiful animations of the fields are also available online: https://www.remcom.com/xfdtd-3d-em-simulation-software Finite Difference Method, Yee Algorithm 427 Figure 37.3: The 2D FDTD simulation of complicated optical waveguides (courtesy of Math- works). Figure 37.4: FDTD simulation of human head in a squirrel cage of an MRI (magnetic reso- nance imaging) system (courtesy of REMCOM). 37.2.1 Finite-Difference Frequency Domain Method Unlike electrical engineering, in many fields, nonlinear problems are prevalent. But when we have a linear time-invariant problem, it is simpler to solve the problem in the frequency domain. This is analogous to perform a time Fourier transform of the pertinent linear equa- tions. Consequently, one can write (37.2.1) to (37.2.6) in the frequency domain to remove the time derivatives. Then one can apply the finite difference approximation to the space deriva- tives using the Yee grid. As a result, one arrives at a matrix equation A · x = b (37.2.11) where x is an unknown vector containing E and H fields, and b is a source vector that drives the system containing J. Due to the near-neighbor interactions of the fields on the Yee 428 Electromagnetic Field Theory grid, the matrix A is highly sparse and contains O(N) non-zero elements. When an iterative method is used to solve the above equation, the major cost is in performing a matrix-vector product A · x. However, in practice, the matrix A is never generated nor stored making this a matrix-free method. Because of the simplicity of the Yee algorithm, a code can be easily written to produce the action of A on x. 37.3 Absorbing Boundary Conditions It will not be complete to close this lecture without mentioning absorbing boundary condi- tions. As computer has finite memory, space of infinitely large extent cannot be simulated with finite computer memory. Hence, it is important to design absorbing boundary conditions at the walls of the simulation domain or box, so that waves impinging on it are not reflected. This mimicks the physics of an infinitely large box. This is analogous to experments in microwave engineering. In order to perform experi- ments in an infinite space, such experiments are usually done in an anechoic (non-echoing or non-reflecting) chamber. An anechoic chamber has its walls padded with absorbing materials or microwave absorbers so as to minimize the reflections off its walls (see Figure 37.5). Figure 37.6 shows an acoustic version of anechoic chamber. Figure 37.5: An anechoic chamber for radio frequency. In such an electromagnetically quiet chamber, interference from other RF equipment is minimized (courtesy of Panasonic). Finite Difference Method, Yee Algorithm 429 Figure 37.6: An acoustic anechoic chamber. In such a chamber, even the breast-feeding sound of a baby can be heard clearly (courtesy of AGH University, Poland). By the same token, in order to simulate numerically an infinite box with a finite-size box, absorbing boundary conditions (ABCs) are designed at its walls. The simplest of such ABCs is the impedance boundary condition. (A transmission line terminated with an impedance reflects less than one terminated with an open or a short circuit.) Another simple ABC is to mimick the Sommerfeld radiation condition (much of this is reviewed in [35]).5 A recently invented ABC is the perfectly matched layers (PML) [251]. Also, another similar ABC is the stretched coordinates PML [252]. Figure 37.7 shows simulation results with and without stretched coordinates PMLs on the walls of the simulation domain [253]. Figure 37.7: Simulation of a source on top of a half-space (left) without stretched coordinates PML; and (right) with stretched coordinats PML [253]. 5ABCs are beyond the scope of these lecture notes. 430 Electromagnetic Field Theory Lecture 38 Quantum Theory of Light The quantum theory of the world is the culmination of a series intellectual exercises. It is often termed the intellectual triumph of the twentieth century. One often says that deciphering the laws of nature is like watching two persons play a chess game with rules unbeknownst to us. By watching the moves, we finally have the revelation about the perplexing rules. But we are grateful that, aided by experimental data, these laws of nature are deciphered by our predecessors. It is important to know that with new quantum theory emerges the quantum theory of light. This theory is intimately related to Maxwell’s equations as shall be seen. These new theories spawn the possibility for quantum technologies, one of which is quantum computing. Others are quantum communication, quantum crytography, quantum sensing and many more. 38.1 Historical Background on Quantum Theory That light is a wave has been demonstrated by Newton’s ring phenomenon [254] in the eighteenth century (1717) (see Figure 38.1). In 1801, Thomas Young demonstrated the double slit experiment for light [255] that further confirmed its wave nature (see Figure 38.2). But by the beginning of the 20-th century, one has to accept that light is both a particle, called a photon, carrying a quantum of energy with momentum, as well as a particle endowed with wave-like behavior. This is called wave-particle duality. We shall outline the historical reason for this development. 431 432 Electromagnetic Field Theory Figure 38.1: A Newton’s rings experiment (courtesy of [254]). Figure 38.2: A Young’s double-slit experiment (courtesy of [256]). As mentioned above, quantum theory is a major intellectual achievement of the twentieth century, even though we are still discovering new knowledge in it. Several major experimental findings led to the revelation of quantum theory of nature. In nature, we know that matter is not infinitely divisible. This is vindicated by the atomic theory of John Dalton (1766- 1844) [257]. So fluid is not infinitely divisible: as when water is divided into smaller pieces, Quantum Theory of Light 433 we will eventually arrive at water molecule, H2O, which is the fundamental building block of water. In turns out that electromagnetic energy is not infinitely divisible either. The electromag- netic radiation out of a heated cavity would have a very different spectrum if electromagnetic energy is infinitely divisible. In order to fit experimental observation of radiation from a heated electromagnetic cavity, Max Planck (1900s) [258] proposed that electromagnetic en- ergy comes in packets or is quantized. Each packet of energy or a quantum of energy E is associated with the frequency of electromagnetic wave, namely E = ~ω = ~2πf = hf (38.1.1) where ~ is now known as the Planck constant and ~ = h/2π = 6.626 × 10−34 J·s (Joule- second). Since ~ is very small, this packet of energy is very small unless ω is large. So it is no surprise that the quantization of electromagnetic field is first associated with light, a very high frequency electromagnetic radiation. A red-light photon at a wavelength of 700 nm −19 corresponds to an energy of approximately 2 eV ≈ 3×10 J ≈ 75 kBT , where kBT denotes the thermal energy from thermal law, and kB is Boltzmann’s constant. This is about 25 meV at room temperature.1 A microwave photon has approximately 1 × 10−5 eV ≈ 10−2meV. The second experimental evidence that light is quantized is the photo-electric effect [259]. It was found that matter emitted electrons when light shined on it. First, the light frequency has to correspond to the “resonant” frequency of the atom. Second, the number of electrons emitted is proportional to the number of packets of energy ~ω that the light carries. This was a clear indication that light energy traveled in packets or quanta as posited by Einstein in 1905. This wave-particle duality concept mentioned at the beginning of this section was not new to quantum theory as electrons were known to behave both like a particle and a wave. The particle nature of an electron was confirmed by the measurement of its charge by Millikan in 1913 in his oil-drop experiment. (The double slit experiment for electron was done in 1927 by Davison and Germer, indicating that an electron has a wave nature as well [255].) In 1924, De Broglie [260] suggested that there is a wave associated with an electron with momentum p such that p = ~k (38.1.2) where k = 2π/λ, the wavenumber. All this knowledge gave hint to the quantum theorists of that era to come up with a new way to describe nature. Classically, particles like an electron moves through space obeying Newton’s laws of mo- tion first established in 1687 [261]. The old way of describing particle motion is known as classical mechanics, and the new way of describing particle motion is known as quantum mechanics. Quantum mechanics is very much motivated by a branch of classical mechanics called Hamiltonian mechanics. We will first use Hamiltonian mechanics to study a simple pendulum and connect it with electromagnetic oscillations. 1This is a number ought to be remembered by semi-conductor scientists as the size of the material bandgap with respect to this thermal energy decides if a material is a semi-conductor at room temperature. 434 Electromagnetic Field Theory 38.2 Connecting Electromagnetic Oscillation to Simple Pendulum The theory for quantization of electromagnetic field was started by Dirac in 1927 [3]. In the beginning, it was called quantum electrodynamics (QED) important for understanding particle physics phenomena and light-matter interactions [262]. Later on, it became impor- tant in quantum optics where quantum effects in electromagnetics technologies first emerged. Now, microwave photons are measurable and are important in quantum computers. Hence, quantum effects are important in the microwave regime as well. Maxwell’s equations originally were inspired by experimental findings of Maxwell’s time, and he beautifully put them together using mathematics known during his time. But Maxwell’s equations can also be “derived” using Hamiltonian mechanics and energy conservation. First, electromagnetic theory can be regarded as for describing an infinite set of coupled harmonic oscillators. In one dimension, when a wave propagates on a string, or an electromagnetic wave propagates on a transmission line, they can be regarded as propagating on a set of coupled harmonic oscillators as shown in Figure 38.3. Maxwell’s equations describe the waves travel- ling in 3D space due to the coupling between an infinite set of harmonic oscillators. In fact, methods have been developed to solve Maxwell’s equations using transmission-line-matrix (TLM) method [263], or the partial element equivalent circuit (PEEC) method [180]. In ma- terials, these harmonic oscillators are atoms or molecules, but vacuum they can be thought of as electron-positron pairs (e-p pairs). Electrons are matters, while positrons are anti-matters. Together, in their quiescent state, they form vacuum or “nothingness”. Hence, vacuum can support the propogation of electromagnetic waves through vast distances: we have received light from galaxies many light-years away. Quantum Theory of Light 435 Figure 38.3: Maxwell’s equations describe the coupling of harmonic oscillators in a 3D space. This is similar to waves propagating on a string or a 1D transmission line, or a 2D array of coupled oscillators. The saw-tooth symbol in the figure represents a spring. The cavity modes in electromagnetics are similar to the oscillation of a pendulum in simple harmonic motion. To understand the quantization of electromagnetic field, we start by connecting these cavity-mode oscillations to the oscillations of a simple pendulum. It is to be noted that fundamentally, electromagnetic oscillation exists because of displacement current. Displacement current exists even in vacuum because vacuum is polarizable, namely that D = εE where for vacuum, ε = ε0. Furthermore, displacement current exists because of the ∂D/∂t term in the generalized Ampere’s law added by Maxwell, namely, ∂D ∇ × H = + J (38.2.1) ∂t Together with Faraday’s law that ∂B ∇ × E = − (38.2.2) ∂t (38.2.1) and (38.2.2) together allow for the existence of wave. The coupling between the two equations gives rise to the “springiness” of electromagnetic fields. Wave exists due to the existence of coupled harmonic oscillators, and at a fundamental level, these harmonic oscillators are electron-positron (e-p) pairs. The fact that they are coupled allows waves to propagate through space, and even in vacuum. 436 Electromagnetic Field Theory Figure 38.4: A one-dimensional cavity solution to Maxwell’s equations is one of the simplest way to solve Maxwell’s equations. To make the problem simpler, we can start by looking at a one dimensional cavity formed by two PEC (perfect electric conductor) plates as shown in Figure 38.4. Assume source-free Maxwell’s equations in between the plates and letting E =xE ˆ x, H =yH ˆ y, then (38.2.1) and (38.2.2) become ∂ ∂ H = −ε E (38.2.3) ∂z y ∂t x ∂ ∂ E = −µ H (38.2.4) ∂z x ∂t y The above are similar to the telegrapher’s equations. We can combine them to arrive at ∂2 ∂2 E = µε E (38.2.5) ∂z2 x ∂t2 x There are infinitely many ways to solve the above partial differential equation. But here, we use separation of variables to solve the above by letting Ex(z, t) = E0(t)f(z). Then we arrive at two separate equations that d2E (t) 0 = −ω2E (t) (38.2.6) dt2 l 0 and d2f(z) = −ω2µεf(z) (38.2.7) dz2 l 2 where ωl is the separation constant. There are infinitely many ways to solve the above 2 2 equations which are also eigenvalue equations where ωl and ωl µε are eigenvalues for the first Quantum Theory of Light 437 and the second equations, respectively. The general solution for (38.2.7) is that E0(t) = E0 cos(ωlt + ψ) (38.2.8) 2 In the above, ωl, which is related to the separation constant, is yet indeterminate. To make ωl determinate, we need to impose boundary conditions. A simple way is to impose homogeneous Dirchlet boundary conditions that f(z) = 0 at z = 0 and z = L. This implies that f(z) = sin(klz). In order to satisfy the boundary conditions at z = 0 and z = L, one deduces that lπ k = , l = 1, 2, 3,... (38.2.9) l L Then, ∂2f(z) = −k2f(z) (38.2.10) ∂z2 l 2 2 where kl = ωl µ. Hence, kl = ωl/c, and the above solution can only exist for discrete frequencies or that lπ ω = c, l = 1, 2, 3,... (38.2.11) l L These are the discrete resonant frequencies ωl of the modes of the 1D cavity. The above solutions for Ex(z, t) can be thought of as the collective oscillations of coupled harmonic oscillators forming the modes of the cavity. At the fundamental level, these oscil- lations are oscillators made by electron-positron pairs. But macroscopically, their collective resonances manifest themselves as giving rise to infinitely many electromagnetic cavity modes. The amplitudes of these modes, E0(t) are simple harmonic oscillations. The resonance between two parallel PEC plates is similar to the resonance of a trans- mission line of length L shorted at both ends. One can see that the resonance of a shorted transmission line is similar to the coupling of infnitely many LC tank circuits. To see this, as shown in Figure 38.3, we start with a single LC tank circuit as a simple harmonic oscillator with only one resonant frequency. When two LC tank circuits are coupled to each other, they will have two resonant frequencies. For N of them, they will have N resonant frequencies. For a continuum of them, they will be infinitely many resonant frequencies or modes as indicated by Equation (38.2.9). What is more important is that the resonance of each of these modes is similar to the resonance of a simple pendulum or a simple harmonic oscillator. For a fixed point in space, the field due to this oscillation is similar to the oscillation of a simple pendulum. As we have seen in the Drude-Lorentz-Sommerfeld mode, for a particle of mass m attached to a spring connected to a wall, where the restoring force is like Hooke’s law, the equation of motion of a pendulum by Newton’s law is d2x m + κx = 0 (38.2.12) dt2 438 Electromagnetic Field Theory where κ is the spring constant, and we assume that the oscillator is not driven by an external force, but is in natural or free oscillation. By letting2 −iωt x = x0e (38.2.13) the above becomes 2 −mω x0 + κx0 = 0 (38.2.14) Again, a non-trivial solution is possible only at the resonant frequency of the oscillator or that when ω = ω0 where r κ ω = (38.2.15) 0 m 2 This is the eigensolution of (38.2.12) with eigenvalue ω0. 38.3 Hamiltonian Mechanics Equation (38.2.12) can be derived by Newton’s law but it can also be derived via Hamiltonian mechanics as well. Since Hamiltonian mechanics motivates quantum mechanics, we will look at the Hamiltonian mechanics view of the equation of motion (EOM) of a simple pendulum given by (38.2.12). Hamiltonian mechanics, developed by Hamilton (1805-1865) [264], is motivated by energy conservation [265]. The Hamiltonian H of a system is given by its total energy, namely that H = T + V (38.3.1) where T is the kinetic energy and V is the potential energy of the system. For a simple pendulum, the kinetic energy is given by 1 1 p2 T = mv2 = m2v2 = (38.3.2) 2 2m 2m where p = mv is the momentum of the particle. The potential energy, assuming that the particle is attached to a spring with spring constant κ, is given by 1 1 V = κx2 = mω2x2 (38.3.3) 2 2 0 Hence, the Hamiltonian is given by p2 1 H = T + V = + mω2x2 (38.3.4) 2m 2 0 2For this part of the lecture, we will switch to using exp(−iωt) time convention as is commonly used in optics and physics literatures. Quantum Theory of Light 439 d 3 At any instant of time t, we assume that p(t) = mv(t) = m dt x(t) is independent of x(t). In other words, they can vary independently of each other. But p(t) and x(t) have to time evolve to conserve energy to keep H, the total energy, constant or independent of time. In other words, d dp ∂H dx ∂H H (p(t), x(t)) = 0 = + (38.3.5) dt dt ∂p dt ∂x Therefore, the Hamilton equations of motion are derived to be4 dp ∂H dx ∂H = − , = (38.3.6) dt ∂x dt ∂p From (38.3.4), we gather that ∂H ∂H p = mω2x, = (38.3.7) ∂x 0 ∂p m Applying (38.3.6), we have5 dx p dp = , = −mω2x (38.3.8) dt m dt 0 Combining the two equations in (38.3.8) above, we have d2x m = −mω2x = −κx (38.3.9) dt2 0 which is also derivable by Newton’s law. A typical harmonic oscillator solution to (38.3.9) is x(t) = x0 cos(ω0t + ψ) (38.3.10) dx The corresponding p(t) = m dt is p(t) = −mx0ω0 sin(ω0t + ψ) (38.3.11) Hence 1 1 H = mω2x2 sin2(ω t + ψ) + mω2x2 cos2(ω t + ψ) 2 0 0 0 2 0 0 0 1 = mω2x2 = E (38.3.12) 2 0 0 And the total energy E is a constant of motion (physicists parlance for a time-independent variable), it depends only on the amplitude x0 of the oscillation. 3p(t) and x(t) are termed conjugate variables in many textbooks. 4Note that the Hamilton equations are determined to within a multiplicative constant, because one has not stipulated the connection between space and time, or we have not calibrated our clock [265]. 5We can also calibrate our clock here so that it agrees with our definition of momentum in the ensuing equation. 440 Electromagnetic Field Theory 38.4 Schr¨odingerEquation (1925) Having seen the Hamiltonian mechanics for describing a simple pendulum which is homomor- phic to a cavity resonator, we shall next see the quantum mechanics description of the same simple pendulum: In other words, we will look at the quantum pendulum. To this end, we will invoke Schr¨odingerequation. Schr¨odingerequation cannot be derived just as in the case Maxwell’s equations. It is a wonderful result of a postulate and a guessing game based on experimental observations [71, 72]. Hamiltonian mechanics says that p2 1 H = + mω2x2 = E (38.4.1) 2m 2 0 where E is the total energy of the oscillator, or pendulum. In classical mechanics, the position x of the particle associated with the pendulum is known with great certainty. But in the quantum world, this position x of the quantum particle is uncertain and is fuzzy. As shall be seen later, x is a random variable.6 To build this uncertainty into a quantum harmonic oscillator, we have to look at it from the quantum world. The position of the particle is described by a wave function,7 which makes the location of the particle uncertain. To this end, Schr¨odingerproposed his equation which is a partial differential equation. He was very much motivated by the experimental revelation then that p = ~k from De Broglie and that E = ~ω from Planck’s law and the photo-electric effect. Equation (38.4.1) can be written more suggestively as 2k2 1 ~ + mω2x2 = ω (38.4.2) 2m 2 0 ~ To add more depth to the above equation, one lets the above become an operator equation that operates on a wave function ψ(x, t) so that 2 ∂2 1 ∂ − ~ ψ(x, t) + mω2x2ψ(x, t) = i ψ(x, t) (38.4.3) 2m ∂x2 2 0 ~∂t If the wave function is of the form ψ(x, t) ∼ eikx−iωt (38.4.4) then upon substituting (38.4.4) back into (38.4.3), we retrieve (38.4.2). Equation (38.4.3) is Schr¨odingerequation in one dimension for the quantum version of the simple harmonic oscillator. In Schr¨odinger equation, we can further posit that the wave function has the general form ψ(x, t) = eikx−iωtA(x, t) (38.4.5) 6For lack of a better notation, we will use x to both denote a position in classical mechanics as well as a random variable in quantum theory. 7Since a function is equivalent to a vector, and this wave function describes the state of the quantum system, this is also called a state vector. Quantum Theory of Light 441 where A(x, t) is a slowly varying function of x and t, compared to eikx−iωt.8 In other words, this is the expression for a wave packet. With this wave packet, the ∂2/∂x2 can be again approximated by −k2 in the short-wavelength limit, as has been done in the parax- ial wave approximation. Furthermore, if the signal is assumed to be quasi-monochromatic, then i~∂/∂tψ(x, t) ≈ ~ω, we again retrieve the classical equation in (38.4.2) from (38.4.3). Hence, the classical equation (38.4.2) is a short wavelength, monochromatic approximation of Schr¨odingerequation. However, as we shall see, the solutions to Schr¨odingerequation are not limited to just wave packets described by (38.4.5). In classical mechanics, the position of a particle is described by the variable x, but in the quantum world, the position of a particle x is a random variable. This property needs to be related to the wavefunction that is the solution to Schr¨odingerequation. For this course, we need only to study the one-dimensional Schr¨odinger equation. The above can be converted into eigenvalue problem, just as in waveguide and cavity problems, using separation of variables, by letting9 −iωnt ψ(x, t) = ψn(x)e (38.4.6) By so doing, (38.4.3) becomes an eigenvalue problem 2 d2 1 − ~ + mω2x2 ψ (x) = E ψ (x) (38.4.7) 2m dx2 2 0 n n n where En = ~ωn is the eigenvalue while ψn(x) is the corresponding eigenfunction. The parabolic x2 potential profile is also known as a potential well as it can provide the restoring force to keep the particle bound to the well classically (see Section 38.3 and (38.3.8)). The above equation is also similar to the electromagnetic equation for a dielectric slab waveguide, where the second term is a dielectric profile (mind you, varying in the x direction) that can trap a waveguide mode. Therefore, the potential well is a trap for the particle both in classical mechanics or in wave physics. The above equation (38.4.7) can be solved in closed form in terms of Hermite-Gaussian functions (1864) [266], or that s r r 1 mω0 − mω0 x2 mω0 ψn(x) = e 2~ Hn x (38.4.8) 2nn! π~ ~ where Hn(y) is a Hermite polynomial, and the eigenvalues are found in closed form as 1 E = n + ω (38.4.9) n 2 ~ 0 Here, the eigenfunction or eigenstate ψn(x) is known as the photon number state (or just a number state) of the solution. It corresponds to having n “photons” in the oscillation. If this is conceived as the collective oscillation of the e-p pairs in a cavity, there are n photons 8Recall that this is similar in spirit when we study high frequency solutions of Maxwell’s equations and paraxial wave approximation. 9 Mind you, the following is ωn, not ω0. 442 Electromagnetic Field Theory corresponding to energy of n~ω0 embedded in the collective oscillation. The larger En is, the larger the number of photons there is. However, there is a curious mode at n = 0. This corresponds to no photon, and yet, there is a wave function ψ0(x). This is the zero-point energy state. This state is there even if the system is at its lowest energy state. It is to be noted that in the quantum world, the position x of the pendulum is random. Moreover, this position x(t) is mapped to the amplitude E0(t) of the field. Hence, it is the amplitude of an electromagnetic oscillation that becomes uncertain and fuzzy as shown in Figure 38.5. Figure 38.5: Schematic representation of the randomness of measured electric field. The elec- tric field amplitude maps to the displacement (position) of the quantum harmonic oscillator, which is a random variable (courtesy of Kira and Koch [267]). Quantum Theory of Light 443 Figure 38.6: Plots of the eigensolutions of the quantum harmonic oscillator (courtesy of Wikipedia [268]). 38.5 Some Quantum Interpretations–A Preview Schr¨odingerused this equation with resounding success. He derived a three-dimensional ver- sion of this to study the wave function and eigenvalues of a hydrogen atom. These eigenvalues En for a hydrogen atom agreed well with experimental observations that had eluded scientists for decades. Schr¨odingerdid not actually understand what these wave functions meant. It was Max Born (1926) who gave a physical interpretation of these wave functions. As mentioned before, in the quantum world, a position x is now a random variable. There is a probability distribution function (PDF) associated with this random variable x. This PDF for x is related to the a wave function ψ(x, t), and it is given |ψ(x, t)|2. Then according to probability theory, the probability of finding the particles in the interval10 [x, x + ∆x] is |ψ(x, t)|2∆x. Since |ψ(x, t)|2 is a probability density function (PDF), and it is necessary that ¢ ∞ dx|ψ(x, t)|2 = 1 (38.5.1) −∞ The average value or expectation value of the random variable x is now given by ¢ ∞ dxx|ψ(x, t)|2 = hx(t)i =x ¯(t) (38.5.2) −∞ 10This is the math notation for an interval [, ]. 444 Electromagnetic Field Theory This is not the most ideal notation, since although x is not a function of time, its expectation value with respect to a time-varying function, ψ(x, t), can be time-varying. Notice that in going from (38.4.1) to (38.4.3), or from a classical picture to a quantum picture, we have let the momentum become p, originally a scalar number in the classical world, become a differential operator, namely that11 ∂ p → pˆ = −i (38.5.3) ~∂x The momentum p of a particle now also becomes uncertain and is a random varible: its expectation value is given by12 ¢ ¢ ∞ ∞ ∗ ∗ ∂ dxψ (x, t)ˆpψ(x, t) = −i~ dxψ (x, t) ψ(x, t) = hpˆ(t)i =p ¯(t) (38.5.4) ∞ −∞ ∂x The expectation values of position x and the momentum operatorp ˆ are measurable in the laboratory. Hence, they are also called observables. 38.5.1 Matrix or Operator Representations We have seen in computational electromagnetics that an operator can be projected into a smaller subspace and manifests itself in different representations. Hence, an operator in quan- tum theory can have different representations depending on the space chosen. For instance, given a matrix equation P · x = b (38.5.5) † we can find a unitary operator U with the property U · U = I. Then the above equation can be rewritten as † U · P · x = U · b → U · P · U · U · x = U · b (38.5.6) Then a new equation is obtained such that 0 0 † P · x0 = b0, P = U · P · U , x0 = U · x b0 = U · b (38.5.7) The operators we have encountered thus far in Schr¨odingerequation are in coordinate space representations.13 In coordinate space representation, the momentum operatorp ˆ = ∂ −i~ ∂x , and the variable x can be regarded as a position operator in coordinate space repre- sentation. The operatorp ˆ and x do not commute. In other words, it can be shown that ∂ [ˆp, x] = −i , x = −i (38.5.8) ~∂x ~ In the classical world, [p, x] = 0, but not in the quantum world. In the equation above, we can elevate x to become an operator by lettingx ˆ = xIˆ, where Iˆ is the identity operator. Then 11We useˆto denote a quantum operator. 12This concept of the average of an operator seldom has an analogue in an intro probality course, but it is called the expectation value of an operator in quantum theory. 13Or just coordinate representation. Quantum Theory of Light 445 bothp ˆ andx ˆ are now operators, and are on the same footing. In this manner, we can rewrite equation (38.5.8) above as [ˆp, xˆ] = −i~Iˆ (38.5.9) By performing unitary transformation, it can be shown that the above identity is coordinate independent: it is true in any representation of the operators. It can be shown easily that when two operators share the same set of eigenfunctions, they commute. When two operatorsp ˆ andx ˆ do not commute, it means that the expectation values of quantities associated with the operators, hpˆi and hxˆi, cannot be determined to arbitrary precision simultaneously. For instance,p ˆ andx ˆ correspond to random variables, then the standard deviation of their measurable values, or their expectation values, obey the uncertainty principle relationship that14 ∆p∆x ≥ ~/2 (38.5.10) where ∆p and ∆x are the standard deviation of the random variables p and x. 38.6 Bizarre Nature of the Photon Number States The photon number states are successful in predicting that the collective e-p oscillations are associated with n photons embedded in the energy of the oscillating modes. However, these number states are bizarre: The expectation values of the position of the quantum pendulum associated these states are always zero. To illustrate further, we form the wave function with a photon-number state −iωnt ψ(x, t) = ψn(x)e Previously, since the ψn(x) are eigenfunctions, they are mutually orthogonal and they can be orthonormalized meaning that ¢ ∞ ∗ dxψn(x)ψn0 (x) = δnn0 (38.6.1) −∞ Then one can easily show that the expectation value of the position of the quantum pendulum in a photon number state is ¢ ¢ ∞ ∞ 2 2 hx(t)i =x ¯(t) = dxx|ψ(x, t)| = dxx|ψn(x)| = 0 (38.6.2) −∞ −∞ because the integrand is always odd symmetric. In other words, the expectation value of the position x of the pendulum is always zero. It can also be shown that the expectation value of the momentum operatorp ˆ is also zero for these photon number states. Hence, there are no classical oscillations that resemble them. Therefore, one has to form new wave functions by linear superposing these photon number states into a coherent state. This will be the discussion in the next lecture. 14The proof of this is quite straightforward but is outside the scope of this course. 446 Electromagnetic Field Theory Lecture 39 Quantum Coherent State of Light As mentioned in the previous lecture, the discovery of Schr¨odingerwave equation was a resounding success. When it was discovered by Schr¨odinger,and appplied to a very simple hydrogen atom, its eigensolutions, especially the eigenvalues En coincide beautifully with spectroscopy experiment of the hydrogen atom. Since the electron wavefunctions inside a hydrogen atom does not have a classical analog, less was known about these wavefunctions. But in QED and quantum optics, the wavefunctions have to be connected with classical electromagnetic oscillations. As seen previously, electromagnetic oscillations resemble those of a pendulum. The original eigenstates of the quantum pendulum were the photon number states also called the Fock states. The connection to the classical pendulum was tenuous, but required by the correspondence principle–quantum phenomena resembles classical phenomena in the high energy limit. This connection was finally established by the establishment of the coherent state. 39.1 The Quantum Coherent State We have seen that a photon number states1 of a quantum pendulum do not have a classical correspondence as the average or expectation values of the position and momentum of the pendulum are always zero for all time for this state. Therefore, we have to seek a time- dependent quantum state that has the classical equivalence of a pendulum. This is the coherent state, which is the contribution of many researchers, most notably, Roy Glauber (1925–2018) [269] in 1963, and George Sudarshan (1931–2018) [270]. Glauber was awarded the Nobel prize in 2005. We like to emphasize again that the modes of an electromagnetic cavity oscillation are homomorphic to the oscillation of classical pendulum. Hence, we first connect the oscillation of a quantum pendulum to a classical pendulum. Then we can connect the oscillation of 1In quantum theory, a “state” is synonymous with a state vector or a function. 447 448 Electromagnetic Field Theory a quantum electromagnetic mode to the classical electromagnetic mode and then to the quantum pendulum. 39.1.1 Quantum Harmonic Oscillator Revisited To this end, we revisit the quantum harmonic oscillator or the quantum pendulum with more mathematical depth. Rewriting Schr¨odingerequation as the eigenequation for the photon number state for the quantum harmonic oscillator, we have 2 d2 1 Hψˆ (x) = − ~ + mω2x2 ψ(x) = Eψ(x). (39.1.1) 2m dx2 2 0 where ψ(x) is the eigenfunction, and E is the eigenvalue. The above can be changed into p mω0 a dimensionless form first by dividing ω0, and then let ξ = x be a dimensionless ~ ~ variable. The above then becomes 2 1 d 2 E − 2 + ξ ψ(ξ) = ψ(ξ) (39.1.2) 2 dξ ~ω0 d ˆ ˆ We can defineπ ˆ = −i dξ and ξ = Iξ to rewrite the Hamiltonian as 1 Hˆ = ω (ˆπ2 + ξˆ2) (39.1.3) 2~ 0 Furthermore, the Hamiltonian in (39.1.2) looks almost like A2 − B2, and hence motivates its factorization. To this end, we first show that 1 d 1 d 1 d2 1 d d √ − + ξ √ + ξ = − + ξ2 − ξ − ξ (39.1.4) 2 dξ 2 dξ 2 dξ2 2 dξ dξ It can be shown easily that as operators (meaning that they will act on a function to their right), the last term on the right-hand side is an identity operator, namely that d d ξ − ξ = Iˆ (39.1.5) dξ dξ Therefore 1 d2 1 d 1 d 1 − + ξ2 = √ − + ξ √ + ξ + (39.1.6) 2 dξ2 2 dξ 2 dξ 2 We define the operator 1 d aˆ† = √ − + ξ (39.1.7) 2 dξ The above is the creation, or raising operator and the reason for its name is obviated later. Moreover, we define 1 d aˆ = √ + ξ (39.1.8) 2 dξ Quantum Coherent State of Light 449 which represents the annihilation or lowering operator. With the above definitions of the raising and lowering operators, it is easy to show that by straightforward substitution that a,ˆ aˆ† =a ˆaˆ† − aˆ†aˆ = Iˆ (39.1.9) Therefore, Schr¨odingerequation (39.1.2) for quantum harmonic oscillator can be rewritten more concisely as 1 1 E aˆ†aˆ +a ˆaˆ† ψ = aˆ†aˆ + ψ = ψ (39.1.10) 2 2 ~ω0 In mathematics, a function is analogous to a vector. So ψ is the implicit representation of a vector. The operator 1 aˆ†aˆ + 2 is an implicit2 representation of an operator, and in this case, a differential operator. So in the above, (39.1.10), is analogous to the matrix eigenvalue equation A · x = λx. Consequently, the Hamiltonian operator can be expressed concisely as 1 Hˆ = ω aˆ†aˆ + (39.1.11) ~ 0 2 Equation (39.1.10) above is in implicit math notation. In implicit Dirac notation, it is 1 E aˆ†aˆ + |ψi = |ψi (39.1.12) 2 ~ω0 In the above, ψ(ξ) is a function which is a vector in a functional space. It is denoted as ψ in math notation and |ψi in Dirac notation. This is also known as the “ket”. The conjugate transpose of a vector in Dirac notation is called a “bra” which is denoted as hψ|. Hence, the 3 inner product between two vectors is denoted as hψ1|ψ2i in Dirac notation. If we denote a photon number state by ψn(x) in explicit notation, ψn in math notation or |ψni in Dirac notation, then we have † 1 En 1 aˆ aˆ + |ψni = |ψni = n + |ψni (39.1.13) 2 ~ω0 2 where we have used the fact that En = (n + 1/2)~ω0. Therefore, by comparing terms in the above, we have † aˆ aˆ|ψni = n|ψni (39.1.14) and the operatora ˆ†aˆ is also known as the number operator because of the above. It is often denoted as nˆ =a ˆ†aˆ (39.1.15) 2 P A notation like A · x, we will call implicit, while a notation i,j Aij xj , we will call explicit. 3There is a one-to-one correspondence of Dirac notation to matrix algebra notation. Aˆ|xi ↔ A · x, hx| ↔ † † x hx1|x2i ↔ x1 · x2. 450 Electromagnetic Field Theory † and |ψni is an eigenvector ofn ˆ =a ˆ aˆ operator with eigenvalue n. It can be further shown by direct substitution that √ √ aˆ|ψni = n|ψn−1i ⇔ aˆ|ni = n|n − 1i (39.1.16) √ √ † † aˆ |ψni = n + 1|ψn+1i ⇔ aˆ |ni = n + 1|n + 1i (39.1.17) hence their names as lowering and raising operator.4 39.2 Some Words on Quantum Randomness and Quan- tum Observables We saw previously that in classical mechanics, the conjugate variables p and x are determin- istic variables. But in the quantum world, they become random variables with means and variance. It was quite easy to see that x is a random variable in the quantum world. But the momentum p is elevated to become a differential operatorp ˆ, and it is not clear that it is a random variable anymore. Quantum theory is a lot richer in content than classical theory. Hence, in quantum theory, conjugate variables like p and x are observables endowed with the properties of mean and variance. For them to be endowed with these properties, they are elevated to become quantum operators, which are the representations of these observables. To be meaningful, a quantum state |ψi has to be defined for a quantum system, and these operators representing observables act on the quantum state. Henceforth, we have to extend the concept of the average of a random variable to the “average” of a quantum operator. Now that we know Dirac notation, we can write the expectation value of the operatorp ˆ with respect to a quantum state ψ as p¯ = hpi = hψ|pˆ|ψi (39.2.1) The above is the elevated way of taking the “average” of an operator which is related to the mean of the random variable p. In the above, the two-some, {p,ˆ |ψi} endows the quantum observable p with random properties. Here, p becomes a random variable whose mean is given by the above formula. As mentioned before, Dirac notation is homomorphic to matrix algebra notation. The above is similar to ψ† · P · ψ =p ¯. This quantityp ¯ is always real if P is a Hermitian matrix. Hence, in (39.2.1), the expectation valuep ¯ is always real ifp ˆ is Hermitian. In fact, it can be proved thatp ˆ is Hermitian in the function space that it is defined. Furthermore, the variance of the random variable p can be derived from the quantum operatorp ˆ with respect to to a quantum state |ψi. It is defined as 2 2 2 2 σp = hψ|(ˆp − p¯) |ψi = hψ|pˆ |ψi − p¯ (39.2.2) 2 where σp is the standard deviation of the random variable p and σp is its variance [71, 72]. 4The above notation for a vector could appear cryptic or too terse to the uninitiated. To parse it, one can always down-convert from an abstract notation to a more explicit notation. Namely, |ni → |ψni → ψn(ξ). Quantum Coherent State of Light 451 The above implies that the definition of the quantum operators and the quantum states † is not unique. One can define a unitary matrix or operator U such that U · U = I. Then the new quantum state is now given by the unitary transform ψ0 = U · ψ. With this, we can easily show that † † p¯ = ψ† · P · ψ = ψ† · U · U · P · U · U · ψ 0 = ψ0† · P · ψ0 (39.2.3) 0 † 0 where P = U·P·U via unitary transform. Now, P is the new quantum operator representing the observable p and ψ0 is the new quantum state vector. In the previous section, we have elevated the position variable or observable ξ to become an operator ξˆ = ξIˆ. This operator is clearly Hermitian, and hence, the expectation value of this position operator is always real. Here, ξˆ is diagonal in the coordinate space representation, but it need not be in other Hilbert space representations using unitary transformation shown above. 39.3 Derivation of the Coherent States As one cannot see the characteristics of a classical pendulum emerging from the photon number states, one needs another way of bridging the quantum world with the classical world. This is the role of the coherent state: It will show the correspondence principle, with a classical pendulum emerging from a quantum pendulum when the energy of the pendulum is large. Hence, it will be interesting to see how the coherent state is derived. The derivation of the coherent state is more math than physics. Nevertheless, the deriva- tion is interesting. We are going to present it according to the simplest way presented in the literature. There are deeper mathematical methods to derive this coherent state like Bogoliubov transform which is outside the scope of this course. Now, endowed with the needed mathematical tools, we can derive the coherent state simply. To say succinctly, the coherent state is the eigenstate of the annihilation operator, namely that aˆ|αi = α|αi (39.3.1) Here, we use α as an eigenvalue as well as an index or identifier of the state |αi.5 Since the number state |ni is complete, the coherent state |αi can be expanded in terms of the number state |ni. Or that ∞ X |αi = Cn|ni (39.3.2) n=0 When the annihilation operator is applied to the above, we have ∞ ∞ ∞ ∞ X X X √ X √ aˆ|αi = Cnaˆ|ni = Cnaˆ|ni = Cn n|n − 1i = Cn+1 n + 1|ni (39.3.3) n=0 n=1 n=1 n=0 5 This notation is cryptic and terse, but one can always down-convert it as |αi → |fαi → fα(ξ) to get a more explicit notation. 452 Electromagnetic Field Theory The last equality follows from changing the variable of summation from n to n + 1. Equating the above with α|αi on the right-hand side of (39.3.1), then ∞ ∞ X √ X Cn+1 n + 1|ni = α Cn|ni (39.3.4) n=0 n=0 By the orthonormality of the number states |ni and the completeness of the set, √ Cn+1 = αCn/ n + 1 (39.3.5) Or recursively √ √ 2 p n Cn = Cn−1α/ n = Cn−2α / n(n − 1) = ... = C0α / n! (39.3.6) Consequently, the coherent state |αi is ∞ X αn |αi = C0 √ |ni (39.3.7) n=0 n! But due to the probabilistic interpretation of quantum mechanics, the state vector |αi is normalized to one, or that6 hα|αi = 1 (39.3.8) Then ∞ n n0 ∗ X α α 0 hα|αi = C C0 √ √ hn |ni 0 0 n,n0 n! n ! ∞ 2n X |α| 2 = |C |2 = |C |2e|α| = 1 (39.3.9) 0 n! 0 n=0 −|α|2/2 Therefore, C0 = e for normalization, or that ∞ n 2 X α |αi = e−|α| /2 √ |ni (39.3.10) n=0 n! In the above, to reduce the double summations into a single summation, we have made use 0 of hn |ni = δn0n, or that the photon-number states are orthonormal. Also sincea ˆ is not a Hermitian operator, its eigenvalue α can be a complex number. Since the coherent state is a linear superposition of the photon number states, an average number of photons can be associated with the coherent state. If the average number of photons embedded in a coherent is N, then it can be shown that N = |α|2. As shall be shown, α is related to the amplitude of the quantum oscillation: The more photons there are in a coherent state, the larger |α| is. ¡ 6 ∞ ∗ The expression can be written more explicitly as hα|αi = hfα|fαi = −∞ dξfα(ξ)fα(ξ) = 1. Quantum Coherent State of Light 453 39.3.1 Time Evolution of a Quantum State The Schr¨odinger equation can be written concisely as ˆ H|ψi = i~∂t|ψi (39.3.11) The above not only entails the form of Schr¨odingerequation, it is the form of the general quantum state equation. Since Hˆ is time independent, the formal solution to the above is ˆ |ψ(t)i = e−iHt/~|ψ(0)i (39.3.12) Meaning of the Function of an Operator At this juncture, it is prudent to digress to discuss the meaning of a function of an operator, which occurs in (39.3.12): The exponential function is a function of the operator Hˆ . This is best explained by expanding the pertinent function into a Taylor series, namely, 1 2 1 n f(A) = f(0)I + f 0(0)A + f 00(0)A + ... + f (n)(0)A + ... (39.3.13) 2! n! Without loss of generality, we have used the matrix operator A as an illustration. The above series has no meaning unless it acts on an eigenvector of the matrix operator A, where Av = λv. Hence, by applying the above equation (39.3.13) to an eigenvector v of A, we have 1 2 1 n f(A)v = f(0)v + f 0(0)Av + f 00(0)A v + ... + f (n)(0)A v + ... 2! n! 1 2 1 = f(0)v + f 0(0)λv + f 00(0)λ v + ... + f (n)(0)λnv + ... = f(λ)v (39.3.14) 2! n! The last equality follows by re-summing the Taylor series back into a function. Applying this to an exponential function of an operator, we have eAv = eλv (39.3.15) Applying this to the photon number state with Hˆ being that of the quantum pendulum and that |ni is the eigenvector of Hˆ , then ˆ e−iHt/~|ni = e−iωnt|ni (39.3.16) 1 ˆ where ωn = n + 2 ω0. In particular, |ni is an eigenstate of the Hamiltonian H for the quantum pendulum, or that from (39.1.11) and (39.1.14) 1 Hˆ |ni = ω |ni = ω n + |ni (39.3.17) ~ n ~ 0 2 In other words, |ni, a shorthand notation for |ψni in (39.1.13), is an eigenvector of Hˆ . 454 Electromagnetic Field Theory Time Evolution of the Coherent State Using the above time-evolution operator, then the time dependent coherent state evolves in time as7 ∞ n −iω t ˆ 2 X α e n |α, ti = e−iHt/~|αi = e−|α| /2 √ |ni (39.3.18) n=0 n! 1 By letting ωn = ω0 n + 2 , the above can be written as n ∞ −iω0t 2 X αe |α, ti = e−iω0t/2e−|α| /2 √ |ni (39.3.19) n=0 n! Now we see that the last factor in (39.3.19) is similar to the expression for a coherent state in (39.3.10). Therefore, we can express the above more succinctly by replacing α in (39.3.10) withα ˜ = αe−iω0t as |α, ti = e−iω0t/2|αe−iω0ti = e−iω0t/2|α˜i (39.3.20) Consequently, aˆ|α, ti =ae ˆ −iω0t/2|αe−iω0ti = e−iω0t/2 αe−iω0t |αe−iω0ti =α ˜|α, ti (39.3.21) Therefore, |α, ti is the eigenfunction of thea ˆ operator with eigenvalueα ˜. But now, the eigenvalue of the annihilation operatora ˆ is a complex number which is a function of time t. It is to be noted that in the coherent state in (39.3.19), the photon number states time-evolve coherently together in a manner to result in a phase shift e−iω0t in the eigenvalue giving rise to a new eigenvalueα ˜! 39.4 More on the Creation and Annihilation Operator As seen in the photon-number states, the oscillation of the pendulum does not emerge in the quantum solutions to Schr¨odingerequation. Hence it is prudent to see if this physical phenomenon emerge with the coherent state. In order to connect the quantum pendulum to a classical pendulum via the coherent state, we will introduce some new operators. Since 1 d aˆ† = √ − + ξ (39.4.1) 2 dξ 1 d aˆ = √ + ξ (39.4.2) 2 dξ We can relatea ˆ† anda ˆ, which are non-hermitian, to the momentum operatorπ ˆ and position operator ξˆ previously defined which are hermitian. Then 1 aˆ† = √ −iπˆ + ξˆ (39.4.3) 2 1 aˆ = √ iπˆ + ξˆ (39.4.4) 2 7 Note that |α, ti is a shorthand for fα(ξ, t). Quantum Coherent State of Light 455 From the above, by subtracting and adding the two equations, we arrive at 1 ξˆ = √ aˆ† +a ˆ = ξIˆ (39.4.5) 2 i d πˆ = √ aˆ† − aˆ = −i (39.4.6) 2 dξ Notice that both ξˆ andπ ˆ are Hermitian operators in the above, with real expectation values in accordance to (39.2.1). With this, the average or expectation value of the position of the pendulum in normalized coordinate, ξ, can be found by taking expectation with respect to the coherent state, or 1 hα|ξˆ|αi = √ hα|aˆ† +a ˆ|αi (39.4.7) 2 Since by taking the complex conjugate transpose of (39.3.1)8 hα|aˆ† = hα|α∗ (39.4.8) and (39.4.7) becomes 1 √ ξ¯ = hξi = hα|ξˆ|αi = √ (α∗ + α) hα|αi = 2 Repeating the exercise for time-dependent case, when we let α → α˜(t) = αe−iω0t, then, letting α = |α|e−iψ yields 1 √ ξ¯(t) = hξ(t)i = hα˜(t)|ξˆ|α˜(t)i = √ (˜α∗(t) +α ˜(t)) hα˜(t)|α˜(t)i = 2 In the above, we use ξ to denote the random variable. So hξ(t)i refers to the average of the random variable ξ, or ξ¯(t) that is a function of time. By the same token, i √ π¯ = hπi = hα|πˆ|αi = √ (α∗ − α) hα|αi = 2=m (α) 6= 0 (39.4.12) 2 For the time-dependent case, we let α → α˜(t) = αe−iω0t, √ π¯(t) = hπ(t)i = − 2|α| sin(ω0t + ψ) (39.4.13) Hence, we see that the expectation values of the normalized coordinate and momentum just behave like a classical pendulum. There is however a marked difference: These values have 8Dirac notation is homomorphic with matrix algebra notation. (a · x)† = x† · (a)†. 456 Electromagnetic Field Theory standard deviations or variances that are non-zero. Thus, they have quantum fluctuation or quantum noise associated with them. Since the quantum pendulum is homomorphic with the oscillation of a quantum electromagnetic mode, the amplitude of a quantum electromagnetic mode will have a mean and a fluctuation as well. Figure 39.1: The time evolution of the coherent state. It follows the motion of a classical pendulum or harmonic oscillator (courtesy of Gerry and Knight [271]). . Figure 39.2: The time evolution of the coherent state for different α’s. The left figure is for α = 5 while the right figure is for α = 10. Recall that N = |α|2. . Quantum Coherent State of Light 457 39.4.1 Connecting Quantum Pendulum to Electromagnetic Oscilla- tor9 We see that the electromagnetic oscillator in a cavity is similar or homomorphic to a pendu- lum. The classical Hamiltonian is p2 1 1 H = T + V = + mω2x2 = P 2(t) + Q2(t) = E (39.4.14) 2m 2 0 2 where E is the total energy of the system. In the above, ω0 is the resonant frequency of the classical pendulum. To make the cavity mode homomorphic to a pendulum, we have to replace ω0 with ωl, the resonant frequency of the cavity mode. In the above, P is a normalized momentum and Q is a normalized coordinate, and their squares have the unit of energy. We have also shown that when the classical pendulum is elevated to be a quantum pendulum, ˆ ˆ † 1 then H → H, where H = ~ωl aˆ aˆ + 2 . Then Schr¨odingerequation becomes 1 Hˆ |ψ, ti = ω aˆ†aˆ + |ψ, ti = i ∂ |ψ, ti (39.4.15) ~ l 2 ~ t Our next task is to connect the electromagnetic oscillator to this pendulum. In general, the total energy or the Hamiltonian of an electromagnetic system is ¢ 1 1 H = dr εE2(r, t) + B2(r, t) . (39.4.16) 2 µ V It is customary to write this Hamiltonian in terms of scalar and vector potentials. For simplicity, we use a 1D cavity, and let A =xA ˆ x, ∇ · A = 0 so that ∂xAx = 0, and letting . Φ = 0. Then B = ∇ × A and E = −A, and the classical Hamiltonian from (39.4.16) for a Maxwellian system becomes ¢ 1 . 1 H = dr εA2(r, t) + (∇ × A(r, t))2 . (39.4.17) 2 µ V . For the 1D case, the above implies that By = ∂zAx, and Ex = −∂tAx = −Ax. Hence, we let Ax = A0(t) sin(klz) (39.4.18) . Ex = −A0(t) sin(klz) = E0(t) sin(klz) (39.4.19) By = klA0(t) cos(klz). (39.4.20) ¡ ¡ . L where E0(t) = −A0(t). After integrating over the volume such that V dr = A 0 dz, the Hamiltonian (39.4.17) then becomes V ε . 2 V H = 0 A (t) + 0 k2A2(t). (39.4.21) 4 0 4µ l 0 9May be skipped on first reading. 458 Electromagnetic Field Theory where V0 = AL, is the mode volume. The form of (39.4.21) now resembles the pendulum Hamiltonian. We can think of A0(t) as being related to the displacement of the pendulum. Hence, the second term resembles the potential energy. The first term has the time derivative of A0(t), and hence, can be connected to the kinetic energy of the system. Therefore, we can rewrite the Hamiltonian as 1 H = P 2(t) + Q2(t) (39.4.22) 2 where r r s V ε . V ε V P (t) = 0 A (t) = − 0 E (t),Q(t) = 0 k A (t) (39.4.23) 2 0 2 0 2µ l 0 By elevating P and Q to be quantum operators, ˆ p ˆ p ˆ P (t) → P = ~ωlπˆ(t),Q(t) → Q = ~ωlξ(t) (39.4.24) so that the quantum Hamiltonian now is 1 h i 1 Hˆ = Pˆ2 + Qˆ2 = ω πˆ2 + ξˆ2 (39.4.25) 2 2~ l similar to (39.1.3) as before except now that the resonant frequency of this mode is ωl instead of ω0 because these are the cavity modes, each of which is homomorphic to a quantum pendulum of frequency ωl. An equation of motion for the state of the quantum system can be associated with the quantum Hamiltonian just as in the quantum pendulum case. We have shown previously that √ aˆ† +a ˆ = 2ξˆ (39.4.26) √ aˆ† − aˆ = − 2iπˆ (39.4.27) Then we can let r V0ε p Pˆ = − Eˆ = ω πˆ (39.4.28) 2 0 ~ l Finally, we arrive at r r 2~ωl 1 ~ωl † Eˆ0 = − πˆ = aˆ − aˆ (39.4.29) εV0 i εV0 Now that E0 has been elevated to be a quantum operator Eˆ0, from (39.4.19), we can put in the space dependence in accordance to (39.4.19) to get Eˆx(z) = Eˆ0 sin(klz) (39.4.30) Consequently, the fully quantized field is a field operator given by r 1 ~ωl † Eˆx(z) = aˆ − aˆ sin(klz) (39.4.31) i εV0 Quantum Coherent State of Light 459 Notice that in the above, Eˆ0, and Eˆx(z) are all Hermitian operators and they correspond to quantum observables that have randomness associated with them but with real mean values. Also, the operators are independent of time because they are in the Schr¨odingerpicture. Heisenberg Picture and Schr¨odingerPicture Because the quantum state equation can be solved formally in closed form, we can look at the quantum world in two pictures: The Heisenberg picture or the Schr¨odingerpicture. A quantum observable is an operator, and the mean value of the observable, which is measurable, is given by O¯(t) = hψ(t)|Oˆs|ψ(t)i. (39.4.32) Alternatively, one can rewrite the above using (39.3.12) as iHt/ˆ −iHt/ˆ O¯(t) = hψ(0)|e ~Oˆse ~|ψ(0)i. (39.4.33) In other words, the above can be rewritten as O¯(t) = hψ(0)|Oˆh(t)|ψ(0)i. (39.4.34) where iHt/ˆ −iHt/ˆ Oˆh(t) = e ~Oˆse ~. (39.4.35) The above illustrates the two different ways to look at the quantum world that produce the same expectation value of a quantum operator. The first way, called the Schr¨odinger picture, keeps the quantum operator Oˆs to be time independent, but retain the quantum state vector |ψ(t)i as a function of time. The second way, called the Heisenberg picture, lets the operator be time dependent, but the state vector is independent of time. Hence, one can derive the equation of motion of an operator in the Heisenberg picture. Also, note that the operators Oˆh and Oˆs in the Heisenberg picture and the Schr¨odingerpicture, respectively, ˆ are related to each other by a unitary transform via the time-evolution operator e−iHt/~. In the Schr¨odingerpicture, to get time dependence fields, one has to take the expectation value of the operators with respect to time-varying quantum state vector like the time-varying coherent state. To let Eˆx have any meaning, it should act on a quantum state. For example, |ψEi = Eˆx|ψi (39.4.36) Notice that thus far, all the operators derived are independent of time. To derive time dependence of these operators, one needs to find their expectation value with respect to time-dependent state vectors.10 To illustrate this, we can take expectation value of the quantum operator Eˆx(z) with 10This is known as the Schr¨odingerpicture. 460 Electromagnetic Field Theory respect to a time dependent state vector, like the time-dependent coherent state, Thus r 1 ~ωl † hEx(z, t)i = hα, t|Eˆx(z)|α, ti = hα, t|aˆ − aˆ|α, ti i εV0 1r ω r ω = ~ l (˜α∗(t) − α˜(t)) hα, t|α, ti = −2 ~ l =m(˜α) (39.4.37) i εV0 εV0 From (39.3.19), we can show that hα, t|α, ti = 1. Using the time-dependentα ˜(t) = αe−iωlt = |α|e−i(ωlt+ψ) in the above, similar to (39.4.13), we have r ~ωl hEx(z, t)i = 2 |α| sin(ωlt + ψ) (39.4.38) εV0 whereα ˜(t) = αe−iωlt. The expectation value of the operator with respect to a time-varying quantum state, in fact, gives rise to a time-varying quantity. The above, which is the average of a random field, resembles a classical field. But since it is rooted in a random variable, it has a standard deviation in addition to having a real-value mean. We can also show that r r 2µ~ωl ˆ µ~ωl † Bˆy(z) = klAˆ0 cos(klz) = ξ = (ˆa +a ˆ) (39.4.39) V0 V0 Again, these are time-independent operators in the Schr¨odingerpicture. To get time-dependent quantities, we have to take the expectation value of the above operator with respect to to a time-varying quantum state. 39.5 Epilogue In conclusion, the quantum theory of light is a rather complex subject. It cannot be taught in just two lectures, but what we wish is to give you a peek into this theory. It takes much longer to learn this subject well: after all, it is the by product of a century of intellectual exercise. This knowledge is still very much in its infancy. Hopefully, the more we teach this subject, the better we can articulate, understand, and explain this subject. 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Index A source on top of a layered medium, 389 Bound electron case horizontal electric dipole, 391 Drude-Lorentz-Sommerfeld model, 85 vertical electric dipole, 390 Boundary conditions, 41 Absorbing boundary conditions, 428 Ampere’s law, 46 Advective equations, 32 conductive media, 52 AM 920 station, 74 Faraday’s law, 44 Ampere’s law, 7, 186 Gauss’s law, 45, 48 Andrew Lloyd Weber Boundary-value problems Cats, 84 BVP, 39 Anisotropic media, 69 Branch cuts, 396 Anisotropic medium, 93 Brewster’s angle, 155 Antenna Brief history net gain, 335 electromagnetics, 6 radiation power density, 335 optics, 6 resonance tunneling, 304 transmit and receive patterns, 335 Cats Aperture antenna, 208 Andrew Lloyd Weber, 84 Array antennas, 293 Cavity resonator, 219 broadside array, 296 cylindrical waveguide resonator, 221 endfire array, 297 full-width half maximum bandwidth, far-field approximation, 294, 298 237 linear phase array, 295 pole location, 236 lobes, 296 quality factor, 233 radiation pattern, 295 quarter wavelength resonator, 221 Axial ratio, 96 transfer function, 236 transmission line model, 219 Back-of-cab formula, 40 Characteristic impedance, 214 Back-of-the-cab formula, 31, 187, 188 Circuit theory, 257 Bell’s theorem, 7 capacitance, 263 Bessel equation, 199 energy storage method, 264 Bessel function, 187, 199, 205 inductor, 262 Bi-anisotropic media, 69 Kirchhoff current law, 257 Biot-Savart law, 50 Kirchhoff voltage law, 257 Born, Max, 443 resistor, 263 477 478 Electromagnetic Field Theory Circular waveguide, 187, 198, 205 Dirichlet boundary condition, 189, 194, cut-off frequency, 200 201 eigenmode, 199 Discrete exterior calculus, 425 Circularly polarized wave, 208 Dispersion relation, 80, 147, 151 Coherent state, 451 plane wave, 75 Compass, 4 Divergence operator, physical meaning, 21 Complex power, 65 Divergence, definition of, 17 Computational electromagnetics, 401 Drude-Lorentz-Sommerfeld model, 82, Conductive media, 69 169, 437 highly conductive case, 80 bound electron case, 85 lowly conductive case, 81 broad applicability of, 86 Conjugate gradient method, 412 damping or dissipation case, 85 Constitutive relations, 29, 69 Duality principle, 140, 149, 162, 212, 214 anisotropic media, 71 bi-anisotropic media, 72 Effective aperture, 336, 338 inhomogeneous media, 72 Effective mass, 88, 89 isotropic frequency dispersive media, Einstein, 2 69 Electric current, 342 nonlinear media, 73 Electric dipole, 142 uniaxial and biaxial media, 72 Electric susceptibility, 30 Constructive interference, 175, 176, 178 Electromagnetic compatibility, 349 Copenhagen school Electromagnetic interference, 349 quantum interpretation, 7 Electromagnetic theory Corpuscular nature of light, 103 unification, 6 Coulomb gauge, 41, 247 Electromagnetic wave, 73 Coulomb’s Law, 8 triumph of Maxwell’s equations, 30 Curl operator, definition, 22 Electromagnetics, 1 Curl operator, physical meaning, 26 history, 1 importance, 1 Damping or dissipation case Electrostatics, 34 Drude-Lorentz-Sommerfeld model, 85 Equivalence theorems, 339 de Broglie, Louis, 433 general case, 341 Derivation inside-out case, 340 Gauss’s law from Coulomb’s law, 12 outside-in case, 341 Diagonalization, 92 Euler’s formula, 62 Dielectric slab, 152 Evanescent wave, 152, 159 Dielectric waveguide, 175 plasma medium, 84 cut-off frequency, 181, 182, 184 Extinction theorem, 341, 345 TE polarization, 178 TM polarization, 183 Far zone, 300 Differential equation, 401 Faraday rotation, 91 Differential forms, 425 Faraday’s law, 7 Dirac, Paul, 2 Fermat’s principle, 363, 392 Directive gain pattern, 276, 289 Finite-Difference Time-Domain, 415 directivity, 290 grid-dispersion error, 422 Quantum Coherent State of Light 479 stability analysis, 420 Gyrotropic medium, 93 Yee algorithm, 424 Folded dipole, 305 Hamilton, William Rowan, 438 Fourier transform technique, 64, 65 Hamiltonian mechanics, 438 Frequency dispersive media, 69, 88 Hankel function, 199, 205 Frequency domain analysis, 61 Heaviside, 28 Fresnel reflection coefficient, 145 Heaviside layer Fresnel zone, 300 Kennelly-Heaviside layer, 84 Fresnel, Austin-Jean, 145 Heinrich Hertz, 271 Functional, 411 experiment, 5 Functional space, 241 Helmholtz equation, 116, 170, 187, 188 Hilbert space, 242 Hermite polynomial, 369 Fundamental mode, 192 Hermite-Gaussian functions, 441 Hermitian matrix, 240 Gauge theory, 2 Hertzian dipole, 271, 301 Gauss’s divergence theorem, 17, 19 far field, 276 Gauss’s Law, 11 near field, 275 Gauss’s law point source approximation, 273 differential operator form, 21 radiation resistance, 279 Gaussian beam, 366 Hidden variable theorey, 7 Gauss’s law, 7 Hollow waveguide, 185, 190, 211 Gedanken experiment, 339 cut-off frequency, 192 General reflection coefficient, 122 cut-off mode, 190 Generalized impedance, 123 cut-off wavelength, 192 Generalized reflection coefficient, 137, eigenmode, 191, 194 166, 175 eigenvalue, 191 Generalized transverse resonance eigenvector, 191 condition, 175, 176, 223 TE mode, 186, 190, 212 Geometrical optics, 362 TM mode, 186, 189 Goos-Hanschen shift, 152, 178 Homogeneous solution, 159, 173, 219, 234, Goubau line, 305, 313 235, 318, 321–323 Green’s function Hooke’s law, 437 dyadic, 347 Horn antenna, 208, 307 scalar, 344 Huygens’ principle static, 36 electromagnetic waves case, 346 Green’s function method, 248 scalar waves case, 344 Green’s theorem, 344 Huygens, Christiaan, 343 vector, 347 Hybrid modes, 210 Green, George, 343 Group velocity, 170, 182, 192, 194 Image theory, 351 Guidance condition, 175, 177, 178, 183, electric charges, 352 191 electric dipoles, 352 Guided mode, 173, 213 magnetic charges, 353 Gyrotropic media magnetic dipoles, 353 Faraday rotation, 91 multiple images, 355 480 Electromagnetic Field Theory perfect magnetic conductor surfaces, Marconi, Guglielmo, 271 354 Matrix eigenvalue problem, 191, 240 Impressed current, 56–58, 293, 328, Maxwell, 28 331–334, 342 lifespan, 28 Inhomogeneous media, 69 marriage, 28 Initial value proble, IVP, 323 poems, 28 Integral equation, 402 Maxwell’s equations Intrinsic impedance, 77, 99 accuracy, 2 broad impact, 2 James Clerk Maxwell differential operator form, 26 displacement current, 5 integral form, 7 Jump condition, 41, 43, 48 quantum regime, 2 Jump discontinuity, 43, 45–48 relativistic invariance, 2 valid over vast length scale, 1 Kao, Charles, 176 Yang-Mills theory, 2 Kernel, 403 Maxwell’s theory Kinetic inductance, 212 wave phenomena, 5 Laguere polynomials, 369 Maxwell, James Clerk, 1 associated, 369 Metasurfaces, 365 Laplace’s equation, 36 Microstrip patch antenna, 307 Laplacian operator, 31, 35 Mie scattering, 379 Lateral wave, 152 Mode conversion, 216 Layered media, 166 Mode matching method, 216 LC tank, 437 Mode orthogonality, 240 Leap-frog scheme, 426 resonant cavity modes, 244 Legendre polynomial waveguide modes, 242 associate, 382 Momentum density, 103 Lenz’s law, 262 Mono-chromatic signal, 168 Linear phase array, 295 Linear time-invariant system, 88 Nano-fabrication, 6 Lode stone, 4 Nanoparticles, 89 Lorentz force law, 82, 91 Natural solution, 219, 235, 321, 322 Lorenz gauge, 248 Near zone, 300 Lossless conditions, 105, 106 Negative resistance, 58 Lossy media, 69 Neumann boundary condition, 188, 190, Love’s equivalence theorem, 404 191, 200 Neumann function, 199 Magnetic charge, 142, 249 Nonlinear media, 69 Magnetic current, 142, 249, 343 Normalized generalized impedance, 123 Magnetic dipole, 142 Null space solution, 173, 322 Magnetic monopole, 142 Numerical methods, 401 Magnetic source, 140 Magnetostatics, 39 Oliver Heaviside, 84 Marconi One-way wave equations, 32 experiment, 84 Optical fiber, 152, 210 Quantum Coherent State of Light 481 Optical theorem, 380 complex, 104 Optimization, 410 instantaneous, 56 Poynting’s vector Paraxial wave, 206 complex, 66, 99 Paraxial wave equation, 366 instantaneous, 58, 65 Pendulum, 434 Probability distribution function, 443 Perfect electric conductor, 351 Perfectly matched layers, 429 Quantum electrodynamics, 434 Permeability, 30 Quantum harmonic oscillator, 440, 448 Permittivity, 30 Quantum theory, 2 Phase matching condition, 147, 150, 157, Quasi-optical antennas, 309 175 Quasi-plane wave, 366 Phase shift, 152 Quasi-static electromagnetic theory, 250, Phase velocity, 77, 88, 168, 182, 192, 210 255 Phasor technique, 61, 62, 79, 83 Quasi-TEM mode, 210 phasor technique, 65 Photo-electric effect, 6 Radiation condition; Sommerfeld Photoelectric effect, 433 radiation condition, 325 Photon number state, 441 Radiation fields, 284 Pilot potential, 196, 205 effective aperture, 290 Pilot potential method, 186 effective area, 290 Pilot vector, 187 far-field approximation, 284 Planck radiation law, 6 locally plane wave approximation, Planck, Max, 433 286 Plane waves Rayleigh distance, 299 in lossy conductive media, 79 Rayleigh scattering, 371 uniform, 75 Reaction Plasma medium, 191 Rumsey, 330 cold collisionless, 83 Reactive power, 67 Plasmonics, 89 Reciprocity theorem, 342 Poisson’s equation, 35 conditions, 331 Polarizability, 374 mathematical derivation, 328 Polarization, 94, 102 two-port network, 331 CP, 94 Rectangular cavity, 222 LHCP, 95 Rectangular waveguide, 187, 190, 194, 195 LHEP, 95 cut-off frequency, 194 RHCP, 95 TE mode, 214 RHEP, 95 TM mode, 214 Polarization density, 29 Reduced wave equation, 189 Power flow Reflection coefficient, 122, 135, 136, 145, LHCP, 98 156, 158, 162 LHEP, 98 Reflector antenna, 278 RHCP, 99 Refractive index, 177 RHEP, 99 Relativistic invariance, 2 Poynting’s theorem Resonant solution, 219, 320–324 482 Electromagnetic Field Theory Resonant solution, homogeneous solution, Surface plasmonic mode, 212 322 Surface plasmonic waveguide, 212 Resonator, 219 Retardation, 116 Tangent plane approximations, 362 RFID, 312 TE polarization, 146 Riemann sheets, 396 reflection coefficient, 148 Rumsey reaction theorem, 330 transmission coefficient, 148 Telegrapher’s equations, 118, 122, 214, Scalar potential, 187, 190 215 electrodynamics, 247 frequency domain, 116 statics, 246 time domain, 115 Schr¨odingerequation, 440 Telegraphy Separation of variables, 190, 198 lack of understanding, 5 Shielding, 349 TEM mode, 186 electrostatic, 349 Time-Harmonic fields, 62 relaxation time, 350 TM mode, 194 Shielding by conductive media, 52 TM polarization, 149 Slot antenna, 306 reflection coefficient, 149 Snell’s law, 147, 157, 364 Transmission coefficient, 149 generalized, 365 Toroidal antenna, 142 Sommerfeld identity, 387, 392 Total internal reflection, 150, 178, 181 Sommerfeld radiation condition, 325 Transmission coefficient, 156, 162, 163 Spatially dispersive, 69 Transmission line, 112, 121, 214 Special relativity, 2 capacitive effects, 113 Spectral representations, 383 characteristic impedance, 115, 119 point source, 384 current, 113, 122 Spherical Bessel equation, 382 distributed lumped element model, Spherical Bessel function, 382 113 Spherical Hankel functions, 382 equivalent circuit, 135, 136, 138 Spherical harmonic, 382 frequency analysis, 116 Spherical Neumann function, 382 inductive effects, 113 Spin angular momentum, 102 load, 122 Standing wave, 190, 195 lossy, 118 Static electricity, 4 matched load, 123 Static electromagnetics parasitic circuit elements, 140 differential operator form, 34 stray/parasitic capacitances, 139 integral form, 8 stray/parasitic inductance, 139 Statics time-domain analysis, 113 electric field, 8 voltage, 113 Stationary phase method, 392 Transmission line matrix, 333, 434 Stokes’s theorem, 22 Transverse resonance condition, 174, 175, structured lights, 370 220 Subspace projection, 407 Transverse wave number, 213 Surface current, 340 Trapped wave, 159 Surface plasmon, 158, 173 Traveling wave, 195 Quantum Coherent State of Light 483 Twin-lead transmission line, 305 statics, 246 Voltaic cell Uncertainty principle, 445 telegraphy, 4 Uniqueness theorem, 317, 324, 339, 341–343, 348 Wave equations, 115 conditions, 320 Wave packet, 169 connection to poles of a linear Wave Phenomenon system, 323 frequency domain, 73 radiation from antenna sources, 324 Wave-particle duality, 431 Vector Poisson’s equation, 40 Weyl identity, 385, 392 Vector potential electrodynamics, 247 Yagi-Uda antenna, 306 non-unique, 41 Young, Thomas, 431